├── _config.yml ├── PYTHON ├── Weingarten │ ├── functions1.txt │ ├── functions2.txt │ ├── functions1.pkl │ ├── functions10.pkl │ ├── functions11.pkl │ ├── functions12.pkl │ ├── functions13.pkl │ ├── functions14.pkl │ ├── functions15.pkl │ ├── functions16.pkl │ ├── functions17.pkl │ ├── functions18.pkl │ ├── functions19.pkl │ ├── functions2.pkl │ ├── functions20.pkl │ ├── functions3.pkl │ ├── functions4.pkl │ ├── functions5.pkl │ ├── functions6.pkl │ ├── functions7.pkl │ ├── functions8.pkl │ ├── functions9.pkl │ ├── functions3.txt │ ├── functions4.txt │ ├── pkl2text.py │ ├── functions5.txt │ ├── functions1.html │ ├── functions2.html │ ├── functions3.html │ ├── functions4.html │ ├── weingarten.html │ ├── functions6.txt │ ├── functions5.html │ ├── functions7.txt │ ├── functions6.html │ ├── pkl2html.py │ ├── functions7.html │ ├── functions8.txt │ ├── functions8.html │ ├── functions9.txt │ ├── functions9.html │ ├── functions10.txt │ ├── functions10.html │ ├── functions11.txt │ └── functions11.html ├── __pycache__ │ ├── IHU_source.cpython-37.pyc │ └── WFG_source.cpython-37.pyc ├── example_WFG.py ├── README.md ├── IHU.py ├── examples.py ├── WFG_source.py └── IHU_source.py ├── MATHEMATICA ├── precomputedWG │ ├── functions1.txt │ ├── functions2.txt │ ├── functions3.txt │ ├── functions4.txt │ ├── functions5.txt │ ├── functions6.txt │ ├── functions7.txt │ ├── functions8.txt │ ├── functions9.txt │ ├── functions10.txt │ └── functions11.txt └── README.md ├── rtnifig.png ├── reference.bib ├── gettingstarted_PYTHON.md ├── README.md └── gettingstarted_MATHEMATICA.md /_config.yml: -------------------------------------------------------------------------------- 1 | theme: jekyll-theme-dinky -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions1.txt: -------------------------------------------------------------------------------- 1 | [[[1], 1/n]] -------------------------------------------------------------------------------- /MATHEMATICA/precomputedWG/functions1.txt: -------------------------------------------------------------------------------- 1 | [[[1], 1/n]] -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions2.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1], 1/(n**2 - 1)], [[2], -1/(n**3 - n)]] -------------------------------------------------------------------------------- /rtnifig.png: -------------------------------------------------------------------------------- https://raw.githubusercontent.com/MotohisaFukuda/RTNI/HEAD/rtnifig.png -------------------------------------------------------------------------------- /MATHEMATICA/precomputedWG/functions2.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1], 1/(n**2 - 1)], [[2], -1/(n**3 - n)]] 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-------------------------------------------------------------------------------- /MATHEMATICA/precomputedWG/functions3.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1, 1], (n**2 - 2)/(n**5 - 5*n**3 + 4*n)], [[2, 1], -1/(n**4 - 5*n**2 + 4)], [[3], 2/(n**5 - 5*n**3 + 4*n)]] -------------------------------------------------------------------------------- /PYTHON/example_WFG.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | 3 | from WFG_source import * 4 | 5 | k = 3 6 | display='yes' 7 | record = 'yes' 8 | 9 | # generating Weingarten functions in the folder "Weingarten". 10 | weigartenFunctionGenerator(k,display,record) 11 | 12 | -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions4.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1, 1, 1], (n**4 - 8*n**2 + 6)/(n**8 - 14*n**6 + 49*n**4 - 36*n**2)], [[2, 1, 1], -1/(n**5 - 10*n**3 + 9*n)], [[3, 1], (2*n**2 - 3)/(n**8 - 14*n**6 + 49*n**4 - 36*n**2)], [[2, 2], (n**2 + 6)/(n**8 - 14*n**6 + 49*n**4 - 36*n**2)], [[4], -5/(n**7 - 14*n**5 + 49*n**3 - 36*n)]] -------------------------------------------------------------------------------- /MATHEMATICA/precomputedWG/functions4.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1, 1, 1], (n**4 - 8*n**2 + 6)/(n**8 - 14*n**6 + 49*n**4 - 36*n**2)], [[2, 1, 1], -1/(n**5 - 10*n**3 + 9*n)], [[3, 1], (2*n**2 - 3)/(n**8 - 14*n**6 + 49*n**4 - 36*n**2)], [[2, 2], (n**2 + 6)/(n**8 - 14*n**6 + 49*n**4 - 36*n**2)], [[4], -5/(n**7 - 14*n**5 + 49*n**3 - 36*n)]] -------------------------------------------------------------------------------- /reference.bib: -------------------------------------------------------------------------------- 1 | @article{fukuda2019rtni, 2 | title={RTNI—A symbolic integrator for Haar-random tensor networks}, 3 | author={Fukuda, Motohisa and Koenig, Robert and Nechita, Ion}, 4 | journal={Journal of Physics A: Mathematical and Theoretical}, 5 | volume={52}, 6 | number={42}, 7 | pages={425303}, 8 | year={2019}, 9 | publisher={IOP Publishing} 10 | } 11 | -------------------------------------------------------------------------------- /PYTHON/Weingarten/pkl2text.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | """ 3 | Created on Sun Oct 7 22:12:14 2018 4 | 5 | @author: M 6 | 7 | A Translator form pickle to text. 8 | """ 9 | import pickle 10 | for k in range(0,21,1): 11 | with open("functions{}.pkl".format(k), "rb") as file: 12 | WF = pickle.load(file) 13 | with open("functions{}.txt".format(k), "w") as output: 14 | output.write(str(WF)) 15 | 16 | 17 | -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions5.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1, 1, 1, 1], (n**4 - 20*n**2 + 78)/(n**9 - 30*n**7 + 273*n**5 - 820*n**3 + 576*n)], [[2, 1, 1, 1], -(n**4 - 14*n**2 + 24)/(n**10 - 30*n**8 + 273*n**6 - 820*n**4 + 576*n**2)], [[3, 1, 1], 2/(n**7 - 21*n**5 + 84*n**3 - 64*n)], [[2, 2, 1], (n**2 - 2)/(n**9 - 30*n**7 + 273*n**5 - 820*n**3 + 576*n)], [[4, 1], -(5*n**2 - 24)/(n**10 - 30*n**8 + 273*n**6 - 820*n**4 + 576*n**2)], [[3, 2], -(2*n**2 + 24)/(n**10 - 30*n**8 + 273*n**6 - 820*n**4 + 576*n**2)], [[5], 14/(n**9 - 30*n**7 + 273*n**5 - 820*n**3 + 576*n)]] -------------------------------------------------------------------------------- /MATHEMATICA/precomputedWG/functions5.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1, 1, 1, 1], (n**4 - 20*n**2 + 78)/(n**9 - 30*n**7 + 273*n**5 - 820*n**3 + 576*n)], [[2, 1, 1, 1], -(n**4 - 14*n**2 + 24)/(n**10 - 30*n**8 + 273*n**6 - 820*n**4 + 576*n**2)], [[3, 1, 1], 2/(n**7 - 21*n**5 + 84*n**3 - 64*n)], [[2, 2, 1], (n**2 - 2)/(n**9 - 30*n**7 + 273*n**5 - 820*n**3 + 576*n)], [[4, 1], -(5*n**2 - 24)/(n**10 - 30*n**8 + 273*n**6 - 820*n**4 + 576*n**2)], [[3, 2], -(2*n**2 + 24)/(n**10 - 30*n**8 + 273*n**6 - 820*n**4 + 576*n**2)], [[5], 14/(n**9 - 30*n**7 + 273*n**5 - 820*n**3 + 576*n)]] -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions1.html: -------------------------------------------------------------------------------- 1 |

Weingarten functions for p = 1

Below are the values of the Weingarten function for permutation size p = 1. The input is given as a partition of p. You can also download them as a text file or as a python pickle file. 2 | 3 | \[\operatorname{Wg}([1]) = \frac{1}{n}\] 4 | -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions2.html: -------------------------------------------------------------------------------- 1 |

Weingarten functions for p = 2

Below are the values of the Weingarten function for permutation size p = 2. The input is given as a partition of p. You can also download them as a text file or as a python pickle file. 2 | 3 | \[\operatorname{Wg}([1, 1]) = \frac{1}{n^{2} - 1}\] 4 | \[\operatorname{Wg}([2]) = - \frac{1}{n^{3} - n}\] 5 | -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions3.html: -------------------------------------------------------------------------------- 1 |

Weingarten functions for p = 3

Below are the values of the Weingarten function for permutation size p = 3. The input is given as a partition of p. You can also download them as a text file or as a python pickle file. 2 | 3 | \[\operatorname{Wg}([1, 1, 1]) = \frac{n^{2} - 2}{n^{5} - 5 n^{3} + 4 n}\] 4 | \[\operatorname{Wg}([2, 1]) = - \frac{1}{n^{4} - 5 n^{2} + 4}\] 5 | \[\operatorname{Wg}([3]) = \frac{2}{n^{5} - 5 n^{3} + 4 n}\] 6 | -------------------------------------------------------------------------------- /MATHEMATICA/README.md: -------------------------------------------------------------------------------- 1 | # RTNI - MATHEMATICA files and usage examples 2 | 3 | A few examples for the usage of the RTNI-MATHEMATICA package are provided here. A detailed description of these example can be found in [arXiv:1902.08539](https://arxiv.org/abs/1902.08539). 4 | 5 | * [Examples from the paper](examples_paper.nb) for the usage of RTNI 6 | * [Additional holographic tensor network examples](examples_holographictensornetworks.nb) for the computation of Renyi entropies of holographic tensor networks 7 | * [Additional examples](examples_momentcalculator.nb) for the computation of moments using **MultinomialexpectationvalueHaar** 8 | 9 | 10 | The files in this directory include: 11 | * [RTNI.wl](RTNI.wl) main RTNI MATHEMATICA package 12 | * [precomputedWG](precomputedWG) subdirectory with precomputed Weingarten functions 13 | -------------------------------------------------------------------------------- /PYTHON/README.md: -------------------------------------------------------------------------------- 1 | # RTNI - PYTHON usage and examples 2 | 3 | A few examples for the usage of the RTNI-PYTHON package are provided here. A detailed description of these example can be found in [arXiv:1902.08539](https://arxiv.org/abs/1902.08539). 4 | 5 | * [Examples from the paper](examples.py) for the usage of RTNI 6 | 7 | The following files are included: 8 | 9 | * [IHU_source.py](IHU_source.py) - main python file for the RTNI package 10 | * [WFG_source.py](WFG_source.py) - python file for the precomputation of Weingarten functions 11 | * [example_WFG.py](example_WFG.py) - example file for the usage of WFG_source.py 12 | * [Weingarten/pkl2text.py](Weingarten/pkl2text.py) - program for conversion of pkl files to txt 13 | 14 | In addition, precomputed Weingarten functions are provided in the subdirectory [Weingarten](Weingarten). 15 | -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions4.html: -------------------------------------------------------------------------------- 1 |

Weingarten functions for p = 4

Below are the values of the Weingarten function for permutation size p = 4. The input is given as a partition of p. You can also download them as a text file or as a python pickle file. 2 | 3 | \[\operatorname{Wg}([1, 1, 1, 1]) = \frac{n^{4} - 8 n^{2} + 6}{n^{8} - 14 n^{6} + 49 n^{4} - 36 n^{2}}\] 4 | \[\operatorname{Wg}([2, 1, 1]) = - \frac{1}{n^{5} - 10 n^{3} + 9 n}\] 5 | \[\operatorname{Wg}([3, 1]) = \frac{2 n^{2} - 3}{n^{8} - 14 n^{6} + 49 n^{4} - 36 n^{2}}\] 6 | \[\operatorname{Wg}([2, 2]) = \frac{n^{2} + 6}{n^{8} - 14 n^{6} + 49 n^{4} - 36 n^{2}}\] 7 | \[\operatorname{Wg}([4]) = - \frac{5}{n^{7} - 14 n^{5} + 49 n^{3} - 36 n}\] 8 | -------------------------------------------------------------------------------- /gettingstarted_PYTHON.md: -------------------------------------------------------------------------------- 1 | # RTNI - Setup instructions for PYTHON 2 | 3 | ## Dependencies 4 | 5 | The PYTHON implementation of RTNI has been developed and tested using PYTHON version 3. For symbolic computation, the 6 | * [sympy](https://www.sympy.org/en/index.html) package is used. 7 | The graph visualization routines additionally require the following python3 packages: 8 | * [matplotlib](https://matplotlib.org/) 9 | * [networkx](https://networkx.github.io/) 10 | as well as the library python3-tk. 11 | 12 | 13 | 14 | ## Installation 15 | 16 | The main routines of the RTNI package are contained in the file [IHU_source.py](PYTHON/IHU_source.py). Precomputed Weingarten functions are provided (respectively generated) in the subfolder [Weingarten](PYTHON/Weingarten). 17 | 18 | * Place these files into the same directly. 19 | * Load the RTNI package using the following command 20 | 21 | ```markdown 22 | from IHU_source import * 23 | ``` 24 | 25 | ## Further steps 26 | 27 | * Look at the [list of files and example code for PYTHON](PYTHON/README.md) 28 | 29 | -------------------------------------------------------------------------------- /PYTHON/Weingarten/weingarten.html: -------------------------------------------------------------------------------- 1 |

Weingarten functions

Links to tables of Weingarten functions 2 | 24 | 25 | -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions6.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1, 1, 1, 1, 1], (n**8 - 41*n**6 + 458*n**4 - 1258*n**2 + 240)/(n**14 - 56*n**12 + 1078*n**10 - 8668*n**8 + 28721*n**6 - 35476*n**4 + 14400*n**2)], [[2, 1, 1, 1, 1], -(n**4 - 24*n**2 + 38)/(n**11 - 47*n**9 + 655*n**7 - 2773*n**5 + 3764*n**3 - 1600*n)], [[3, 1, 1, 1], (2*n**6 - 51*n**4 + 229*n**2 - 60)/(n**14 - 56*n**12 + 1078*n**10 - 8668*n**8 + 28721*n**6 - 35476*n**4 + 14400*n**2)], [[2, 2, 1, 1], (n**4 - 3*n**2 + 10)/(n**12 - 40*n**10 + 438*n**8 - 1660*n**6 + 2161*n**4 - 900*n**2)], [[4, 1, 1], -(5*n**2 - 13)/(n**11 - 40*n**9 + 438*n**7 - 1660*n**5 + 2161*n**3 - 900*n)], [[3, 2, 1], -(2*n**2 + 13)/(n**11 - 47*n**9 + 655*n**7 - 2773*n**5 + 3764*n**3 - 1600*n)], [[5, 1], (14*n**2 - 140)/(n**12 - 55*n**10 + 1023*n**8 - 7645*n**6 + 21076*n**4 - 14400*n**2)], [[2, 2, 2], -(n**4 + n**2 + 358)/(n**13 - 56*n**11 + 1078*n**9 - 8668*n**7 + 28721*n**5 - 35476*n**3 + 14400*n)], [[4, 2], (5*n**4 + 75*n**2 + 40)/(n**14 - 56*n**12 + 1078*n**10 - 8668*n**8 + 28721*n**6 - 35476*n**4 + 14400*n**2)], [[3, 3], (4*n**4 + 116*n**2 - 360)/(n**14 - 56*n**12 + 1078*n**10 - 8668*n**8 + 28721*n**6 - 35476*n**4 + 14400*n**2)], [[6], -42/(n**11 - 55*n**9 + 1023*n**7 - 7645*n**5 + 21076*n**3 - 14400*n)]] -------------------------------------------------------------------------------- /MATHEMATICA/precomputedWG/functions6.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1, 1, 1, 1, 1], (n**8 - 41*n**6 + 458*n**4 - 1258*n**2 + 240)/(n**14 - 56*n**12 + 1078*n**10 - 8668*n**8 + 28721*n**6 - 35476*n**4 + 14400*n**2)], [[2, 1, 1, 1, 1], -(n**4 - 24*n**2 + 38)/(n**11 - 47*n**9 + 655*n**7 - 2773*n**5 + 3764*n**3 - 1600*n)], [[3, 1, 1, 1], (2*n**6 - 51*n**4 + 229*n**2 - 60)/(n**14 - 56*n**12 + 1078*n**10 - 8668*n**8 + 28721*n**6 - 35476*n**4 + 14400*n**2)], [[2, 2, 1, 1], (n**4 - 3*n**2 + 10)/(n**12 - 40*n**10 + 438*n**8 - 1660*n**6 + 2161*n**4 - 900*n**2)], [[4, 1, 1], -(5*n**2 - 13)/(n**11 - 40*n**9 + 438*n**7 - 1660*n**5 + 2161*n**3 - 900*n)], [[3, 2, 1], -(2*n**2 + 13)/(n**11 - 47*n**9 + 655*n**7 - 2773*n**5 + 3764*n**3 - 1600*n)], [[5, 1], (14*n**2 - 140)/(n**12 - 55*n**10 + 1023*n**8 - 7645*n**6 + 21076*n**4 - 14400*n**2)], [[2, 2, 2], -(n**4 + n**2 + 358)/(n**13 - 56*n**11 + 1078*n**9 - 8668*n**7 + 28721*n**5 - 35476*n**3 + 14400*n)], [[4, 2], (5*n**4 + 75*n**2 + 40)/(n**14 - 56*n**12 + 1078*n**10 - 8668*n**8 + 28721*n**6 - 35476*n**4 + 14400*n**2)], [[3, 3], (4*n**4 + 116*n**2 - 360)/(n**14 - 56*n**12 + 1078*n**10 - 8668*n**8 + 28721*n**6 - 35476*n**4 + 14400*n**2)], [[6], -42/(n**11 - 55*n**9 + 1023*n**7 - 7645*n**5 + 21076*n**3 - 14400*n)]] -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions5.html: -------------------------------------------------------------------------------- 1 |

Weingarten functions for p = 5

Below are the values of the Weingarten function for permutation size p = 5. The input is given as a partition of p. You can also download them as a text file or as a python pickle file. 2 | 3 | \[\operatorname{Wg}([1, 1, 1, 1, 1]) = \frac{n^{4} - 20 n^{2} + 78}{n^{9} - 30 n^{7} + 273 n^{5} - 820 n^{3} + 576 n}\] 4 | \[\operatorname{Wg}([2, 1, 1, 1]) = \frac{- n^{4} + 14 n^{2} - 24}{n^{10} - 30 n^{8} + 273 n^{6} - 820 n^{4} + 576 n^{2}}\] 5 | \[\operatorname{Wg}([3, 1, 1]) = \frac{2}{n^{7} - 21 n^{5} + 84 n^{3} - 64 n}\] 6 | \[\operatorname{Wg}([2, 2, 1]) = \frac{n^{2} - 2}{n^{9} - 30 n^{7} + 273 n^{5} - 820 n^{3} + 576 n}\] 7 | \[\operatorname{Wg}([4, 1]) = \frac{- 5 n^{2} + 24}{n^{10} - 30 n^{8} + 273 n^{6} - 820 n^{4} + 576 n^{2}}\] 8 | \[\operatorname{Wg}([3, 2]) = \frac{- 2 n^{2} - 24}{n^{10} - 30 n^{8} + 273 n^{6} - 820 n^{4} + 576 n^{2}}\] 9 | \[\operatorname{Wg}([5]) = \frac{14}{n^{9} - 30 n^{7} + 273 n^{5} - 820 n^{3} + 576 n}\] 10 | -------------------------------------------------------------------------------- /PYTHON/IHU.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | """ 3 | Created on Sat Aug 18 22:36:03 2018 4 | 5 | @author: M 6 | """ 7 | from IHU_source import * 8 | 9 | 10 | 11 | ##########Tr[(id \otimes Tr) A] aftere########## 12 | 13 | d,k,n = symbols('d k n') 14 | 15 | e1 = [["A" , 1 , "out" , 1] , ["U" , 1 , "in" , 1]] 16 | e2 = [["A" , 1 , "out" , 2] , ["U" , 1 , "in" , 2]] 17 | e3 = [["U*" , 1 , "out" , 1] , ["A" , 1 , "in" , 1]] 18 | e4 = [["U*" , 1 , "out" , 2] , ["A" , 1 , "in" , 2]] 19 | e5 = [["U" , 1 , "out" , 2] , ["U*" , 1 , "in" , 2]] 20 | g=[e1,e2,e3,e4,e5] 21 | gw=[g,1] 22 | visualizeTN(gw) 23 | rm = ["U",[n,k],[n,k],n*k] 24 | Eg = integrateHaarUnitary(gw,rm) 25 | print(Eg) 26 | visualizeTN(Eg) 27 | 28 | ########## 29 | 30 | 31 | 32 | 33 | 34 | """ 35 | 36 | e1 = [["A", 1, "R", 1], ["B", 1, "L", 1]] 37 | e2 = [["A", 1, "R", 2], ["B", 1, "L", 2]] 38 | e3 = [["A", 2, "R", 1], ["B", 2, "L", 1]] 39 | e4 = [["A", 2, "R", 2], ["B", 2, "L", 2]] 40 | g1 = [e1, e2, e3, e4] 41 | 42 | f2 = [["A", 1, "R", 2], ["B", 2, "L", 2]] 43 | f4 = [["A", 2, "R", 2], ["B", 1, "L", 2]] 44 | g2 = [e1, f2, e3, f4] 45 | 46 | h1 = [["A", 1, "R", 1], ["B", 2, "L", 1]] 47 | h2 = [["A", 1, "R", 2], ["B", 2, "L", 2]] 48 | h3 = [["A", 2, "R", 1], ["B", 1, "L", 1]] 49 | h4 = [["A", 2, "R", 2], ["B", 1, "L", 2]] 50 | g3 = [h1,h2,h3,h4] 51 | 52 | k1 = [["A", 1, "R", 1], ["B", 3, "L", 1]] 53 | k2 = [["A", 1, "R", 2], ["B", 3, "L", 2]] 54 | k3 = [["A", 2, "R", 1], ["B", 1, "L", 1]] 55 | k4 = [["A", 2, "R", 2], ["B", 1, "L", 2]] 56 | g4 = [k1,k2,k3,k4] 57 | 58 | 59 | gwA=[[g1,1],[g2,1]] 60 | gwB=[[g1,1],[g3,1]] 61 | gwC=[[g1,1],[g4,1]] 62 | rm = [] 63 | EgA = integrateHaarUnitary(gwA,rm) 64 | EgB = integrateHaarUnitary(gwB,rm) 65 | EgC = integrateHaarUnitary(gwC,rm) 66 | print(EgA) 67 | print(EgB) 68 | print(EgC) 69 | #visualizeTN(Eg) 70 | 71 | 72 | """ 73 | 74 | 75 | -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions7.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1, 1, 1, 1, 1, 1], (n**8 - 71*n**6 + 1568*n**4 - 11398*n**2 + 15780)/(n*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[2, 1, 1, 1, 1, 1], (-n**8 + 61*n**6 - 1058*n**4 + 4958*n**2 - 1440)/(n**2*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[3, 1, 1, 1, 1], 2*(n**6 - 51*n**4 + 644*n**2 - 1434)/(n*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[2, 2, 1, 1, 1], (n**6 - 45*n**4 + 488*n**2 - 1284)/(n*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[4, 1, 1, 1], (-5*n**6 + 207*n**4 - 1762*n**2 + 720)/(n**2*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[3, 2, 1, 1], -2/(n**10 - 66*n**8 + 1353*n**6 - 10648*n**4 + 30096*n**2 - 20736)], [[5, 1, 1], 14*(n**2 - 6)/(n*(n**12 - 67*n**10 + 1419*n**8 - 12001*n**6 + 40744*n**4 - 50832*n**2 + 20736))], [[2, 2, 2, 1], (-n**6 + 23*n**4 - 382*n**2 + 2880)/(n**2*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[4, 2, 1], (5*n**2 + 51)/(n*(n**12 - 76*n**10 + 1878*n**8 - 17428*n**6 + 61921*n**4 - 78696*n**2 + 32400))], [[3, 3, 1], 4*(n**4 + 8*n**2 - 429)/(n*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[6, 1], (-42*n**2 + 720)/(n**2*(n**12 - 91*n**10 + 3003*n**8 - 44473*n**6 + 296296*n**4 - 773136*n**2 + 518400))], [[3, 2, 2], 2*(n**4 + 5*n**2 + 834)/(n*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[5, 2], -(14*n**4 + 226*n**2 + 1440)/(n**2*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[4, 3], (-10*n**4 - 470*n**2 + 2160)/(n**2*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[7], 132/(n*(n**12 - 91*n**10 + 3003*n**8 - 44473*n**6 + 296296*n**4 - 773136*n**2 + 518400))]] -------------------------------------------------------------------------------- /MATHEMATICA/precomputedWG/functions7.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1, 1, 1, 1, 1, 1], (n**8 - 71*n**6 + 1568*n**4 - 11398*n**2 + 15780)/(n*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[2, 1, 1, 1, 1, 1], (-n**8 + 61*n**6 - 1058*n**4 + 4958*n**2 - 1440)/(n**2*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[3, 1, 1, 1, 1], 2*(n**6 - 51*n**4 + 644*n**2 - 1434)/(n*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[2, 2, 1, 1, 1], (n**6 - 45*n**4 + 488*n**2 - 1284)/(n*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[4, 1, 1, 1], (-5*n**6 + 207*n**4 - 1762*n**2 + 720)/(n**2*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[3, 2, 1, 1], -2/(n**10 - 66*n**8 + 1353*n**6 - 10648*n**4 + 30096*n**2 - 20736)], [[5, 1, 1], 14*(n**2 - 6)/(n*(n**12 - 67*n**10 + 1419*n**8 - 12001*n**6 + 40744*n**4 - 50832*n**2 + 20736))], [[2, 2, 2, 1], (-n**6 + 23*n**4 - 382*n**2 + 2880)/(n**2*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[4, 2, 1], (5*n**2 + 51)/(n*(n**12 - 76*n**10 + 1878*n**8 - 17428*n**6 + 61921*n**4 - 78696*n**2 + 32400))], [[3, 3, 1], 4*(n**4 + 8*n**2 - 429)/(n*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[6, 1], (-42*n**2 + 720)/(n**2*(n**12 - 91*n**10 + 3003*n**8 - 44473*n**6 + 296296*n**4 - 773136*n**2 + 518400))], [[3, 2, 2], 2*(n**4 + 5*n**2 + 834)/(n*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[5, 2], -(14*n**4 + 226*n**2 + 1440)/(n**2*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[4, 3], (-10*n**4 - 470*n**2 + 2160)/(n**2*(n**14 - 92*n**12 + 3094*n**10 - 47476*n**8 + 340769*n**6 - 1069432*n**4 + 1291536*n**2 - 518400))], [[7], 132/(n*(n**12 - 91*n**10 + 3003*n**8 - 44473*n**6 + 296296*n**4 - 773136*n**2 + 518400))]] -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions6.html: -------------------------------------------------------------------------------- 1 |

Weingarten functions for p = 6

Below are the values of the Weingarten function for permutation size p = 6. The input is given as a partition of p. You can also download them as a text file or as a python pickle file. 2 | 3 | \[\operatorname{Wg}([1, 1, 1, 1, 1, 1]) = \frac{n^{8} - 41 n^{6} + 458 n^{4} - 1258 n^{2} + 240}{n^{14} - 56 n^{12} + 1078 n^{10} - 8668 n^{8} + 28721 n^{6} - 35476 n^{4} + 14400 n^{2}}\] 4 | \[\operatorname{Wg}([2, 1, 1, 1, 1]) = \frac{- n^{4} + 24 n^{2} - 38}{n^{11} - 47 n^{9} + 655 n^{7} - 2773 n^{5} + 3764 n^{3} - 1600 n}\] 5 | \[\operatorname{Wg}([3, 1, 1, 1]) = \frac{2 n^{6} - 51 n^{4} + 229 n^{2} - 60}{n^{14} - 56 n^{12} + 1078 n^{10} - 8668 n^{8} + 28721 n^{6} - 35476 n^{4} + 14400 n^{2}}\] 6 | \[\operatorname{Wg}([2, 2, 1, 1]) = \frac{n^{4} - 3 n^{2} + 10}{n^{12} - 40 n^{10} + 438 n^{8} - 1660 n^{6} + 2161 n^{4} - 900 n^{2}}\] 7 | \[\operatorname{Wg}([4, 1, 1]) = \frac{- 5 n^{2} + 13}{n^{11} - 40 n^{9} + 438 n^{7} - 1660 n^{5} + 2161 n^{3} - 900 n}\] 8 | \[\operatorname{Wg}([3, 2, 1]) = \frac{- 2 n^{2} - 13}{n^{11} - 47 n^{9} + 655 n^{7} - 2773 n^{5} + 3764 n^{3} - 1600 n}\] 9 | \[\operatorname{Wg}([5, 1]) = \frac{14 n^{2} - 140}{n^{12} - 55 n^{10} + 1023 n^{8} - 7645 n^{6} + 21076 n^{4} - 14400 n^{2}}\] 10 | \[\operatorname{Wg}([2, 2, 2]) = \frac{- n^{4} - n^{2} - 358}{n^{13} - 56 n^{11} + 1078 n^{9} - 8668 n^{7} + 28721 n^{5} - 35476 n^{3} + 14400 n}\] 11 | \[\operatorname{Wg}([4, 2]) = \frac{5 n^{4} + 75 n^{2} + 40}{n^{14} - 56 n^{12} + 1078 n^{10} - 8668 n^{8} + 28721 n^{6} - 35476 n^{4} + 14400 n^{2}}\] 12 | \[\operatorname{Wg}([3, 3]) = \frac{4 n^{4} + 116 n^{2} - 360}{n^{14} - 56 n^{12} + 1078 n^{10} - 8668 n^{8} + 28721 n^{6} - 35476 n^{4} + 14400 n^{2}}\] 13 | \[\operatorname{Wg}([6]) = - \frac{42}{n^{11} - 55 n^{9} + 1023 n^{7} - 7645 n^{5} + 21076 n^{3} - 14400 n}\] 14 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # RTNI - A SYMBOLIC INTEGRATOR FOR HAAR-RANDOM TENSOR NETWORKS 2 | 3 | RTNI is symbolic computer algebra package for MATHEMATICA and PYTHON. drawing It computes averages of tensor networks containing multiple Haar-distributed random unitary matrices and symbolic tensors. Such tensor networks are represented as multigraphs, with vertices corresponding to tensors or random unitaries and edges corresponding to tensor contractions. Input and output spaces of random unitaries may be subdivided into arbitrary tensor factors, with dimensions treated symbolically. The algorithm implements the graphical Weingarten calculus and produces a weighted sum of tensor networks representing the average over the unitary group. Associated visualization routines are also provided. 4 | 5 | A detailed description of the functionality of this package with examples of its usage is available at [arXiv:1902.08539](https://arxiv.org/abs/1902.08539). 6 | 7 | ### MATHEMATICA 8 | 9 | * Follow the [RTNI setup guide for MATHEMATICA](gettingstarted_MATHEMATICA.md) to setup RTNI 10 | * Look at the [list of files and sample code for MATHEMATICA](MATHEMATICA/README.md) 11 | 12 | ### PYTHON (A major update was made as [PyRTNI2](https://github.com/MotohisaFukuda/PyRTNI2)) 13 | 14 | * Follow the [RTNI setup guide for PYTHON](gettingstarted_PYTHON.md) to setup RTNI 15 | * Look at the [list of files and sample code for PYTHON](PYTHON/README.md) 16 | * A list of Weingarten functions for permutation size up to 20 can be found [here](https://motohisafukuda.github.io/RTNI/PYTHON/Weingarten/weingarten.html) 17 | 18 | ### ONLINE 19 | 20 | * A light, online version of the code implementing the main integration routine can be found [here](https://github.com/MotohisaFukuda/RTNI_light) 21 | 22 | 23 | ### LICENSE 24 | This project is licensed under the terms of the [GNU GENERAL PUBLIC LICENSE v3.0](LICENSE.txt). When using this software, please include the following reference (also in [BibTeX](reference.bib) format) 25 | 26 | 27 | * Motohisa Fukuda, Robert König, and Ion Nechita. RTNI - A symbolic integrator for Haar-random tensor networks. \[ [doi](https://doi.org/10.1088/1751-8121/ab434b) | Journal of Physics A: Mathematical and Theoretical, Volume 52, Number 42, 2019 \]. 28 | -------------------------------------------------------------------------------- /PYTHON/Weingarten/pkl2html.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | """ 3 | Created on Mon Feb 25 14:50:43 2019 4 | 5 | @author: home 6 | """ 7 | 8 | def latexIt(s): 9 | return sympy.latex(sympy.sympify(s)) 10 | 11 | import os 12 | from yattag import Doc 13 | import pickle 14 | import sympy 15 | 16 | location = os.getcwd() 17 | counter = 0 18 | 19 | 20 | for file in os.listdir(location): 21 | try: 22 | if file.endswith(".pkl"): 23 | print ("pkl file found:\t", file) 24 | counter = counter+1 25 | 26 | doc, tag, text = Doc().tagtext() 27 | doc.asis('') 28 | 29 | # parse input file 30 | wg = pickle.load(open(file,"rb")) 31 | 32 | p = int(file[9:-4]) 33 | 34 | with tag('html'): 35 | with tag('head'): 36 | doc.asis('') 37 | with tag('body'): 38 | with tag('h1'): 39 | text('Weingarten functions for p = ' + str(p)) 40 | text('Below are the values of the ') 41 | with tag('a', ('href','https://en.wikipedia.org/wiki/Weingarten_function')): 42 | text('Weingarten function') 43 | text(' for permutation size p = '+str(p)+'. The input is given as a partition of p. You can also download them as a text ') 44 | with tag('a', ('href',file[:-3]+'txt')): 45 | text('file') 46 | text(' or as a python pickle ') 47 | with tag('a', ('href',file)): 48 | text('file') 49 | text('.\n\n') 50 | for part in wg: 51 | text("\[\operatorname{Wg}("+str(part[0])+") = "+latexIt(str(part[1]))+"\]\n"); 52 | 53 | 54 | # write html in file 55 | fout = open(file[:-3]+"html", "w") 56 | fout.write(doc.getvalue()) 57 | fout.close() 58 | 59 | 60 | except Exception as e: 61 | raise e 62 | print ("No files found!") 63 | 64 | print ("Total pickle files found:\t", counter) 65 | 66 | -------------------------------------------------------------------------------- /gettingstarted_MATHEMATICA.md: -------------------------------------------------------------------------------- 1 | # RTNI - Setup instructions for MATHEMATICA 2 | 3 | ## Dependencies 4 | 5 | The MATHEMATICA implementation of RTNI has been developed and tested using Wolfram Mathematica version 11. 6 | 7 | ## Installation 8 | 9 | The package is contained in the Wolfram Language Package file [RTNI.wl](MATHEMATICA/RTNI.wl). Precomputed Weingarten functions are provided in the subfolder [precomputedWG](MATHEMATICA/precomputedWG). Proceed as follows: 10 | 11 | * Place [RTNI.wl](MATHEMATICA/RTNI.wl) in a directory. 12 | * Place the subfolder [precomputedWG](MATHEMATICA/precomputedWG) in the same directory. 13 | * Open/Create a new mathematica-file in the directory. 14 | * Load the RTNI package using the following commands 15 | 16 | ```markdown 17 | SetDirectory[NotebookDirectory[]]; 18 | << RTNI` 19 | ``` 20 | 21 | If the corresponding output is of the form 22 | ``` 23 | Package RTNI (Random Tensor Network Integrator) version 1.0.5 (last modification: 26/01/2019). 24 | 25 | Loading precomputed Weingarten Functions from /precomputedWG/functions1.txt 26 | Loading precomputed Weingarten Functions from /precomputedWG/functions2.txt 27 | Loading precomputed Weingarten Functions from /precomputedWG/functions3.txt 28 | Loading precomputed Weingarten Functions from /precomputedWG/functions4.txt 29 | Loading precomputed Weingarten Functions from /precomputedWG/functions5.txt 30 | Loading precomputed Weingarten Functions from /precomputedWG/functions6.txt 31 | Loading precomputed Weingarten Functions from /precomputedWG/functions7.txt 32 | Loading precomputed Weingarten Functions from /precomputedWG/functions8.txt 33 | Loading precomputed Weingarten Functions from /precomputedWG/functions9.txt 34 | Loading precomputed Weingarten Functions from /precomputedWG/functions10.txt 35 | Loading precomputed Weingarten Functions from /precomputedWG/functions11.txt 36 | Loading precomputed Weingarten Functions from /precomputedWG/functions12.txt 37 | Loading precomputed Weingarten Functions from /precomputedWG/functions13.txt 38 | Loading precomputed Weingarten Functions from /precomputedWG/functions14.txt 39 | Loading precomputed Weingarten Functions from /precomputedWG/functions15.txt 40 | Loading precomputed Weingarten Functions from /precomputedWG/functions16.txt 41 | Loading precomputed Weingarten Functions from /precomputedWG/functions17.txt 42 | Loading precomputed Weingarten Functions from /precomputedWG/functions18.txt 43 | Loading precomputed Weingarten Functions from /precomputedWG/functions19.txt 44 | Loading precomputed Weingarten Functions from /precomputedWG/functions20.txt 45 | ``` 46 | 47 | then the package and the precomputed functions have been correctly loaded. 48 | 49 | ## Further steps 50 | 51 | * Look at the [List of files and sample code for MATHEMATICA](MATHEMATICA/README.md) 52 | 53 | -------------------------------------------------------------------------------- /PYTHON/examples.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | 3 | 4 | 5 | 6 | ##### 6.1.1 ##### 7 | 8 | from IHU_source import * 9 | 10 | e1 = [["U", 1, "out", 1], ["X", 1, "in", 1]] 11 | e2 = [["Y", 1, "out", 1], ["U", 1, "in", 1]] 12 | e3 = [["U*", 1, "out", 1], ["Y", 1, "in", 1]] 13 | e4 = [["X", 1, "out", 1], ["U*", 1, "in", 1]] 14 | g = [e1, e2, e3, e4] 15 | gw = [g,1] 16 | print(gw) 17 | visualizeTN(gw) 18 | 19 | d = symbols('d') 20 | Eg = integrateHaarUnitary(gw, ["U", [d], [d], d]) 21 | print(Eg) 22 | visualizeTN(Eg) 23 | 24 | 25 | 26 | ##### 6.1.2 ##### 27 | from IHU_source import * 28 | 29 | e1 = [["U", 1, "out", 1], ["X", 1, "in", 1]] 30 | e2 = [["Y", 1, "out", 1], ["U", 1, "in", 1]] 31 | e3 = [["U*", 1, "out", 1], ["Y", 1,"in", 1]] 32 | g = [e1, e2, e3] 33 | gw = [g,1] 34 | 35 | d = symbols('d') 36 | Eg = integrateHaarUnitary(gw, ["U", [d], [d], d]) 37 | print(Eg) 38 | visualizeTN(Eg) 39 | 40 | 41 | 42 | ##### 6.2 ##### 43 | from IHU_source import * 44 | 45 | e1 = [["A", 1, "out", 1], ["U", 1, "in", 1]] 46 | e2 = [["A", 1, "out", 2], ["U", 1, "in", 2]] 47 | e3 = [["U*", 1, "out", 1], ["A", 1, "in", 1]] 48 | e4 = [["U*", 1, "out", 2], ["A", 1, "in", 2]] 49 | e5 = [["U", 1, "out", 2], ["U*", 1, "in", 2]] 50 | g = [e1, e2, e3, e4, e5] 51 | gw = [g,1] 52 | print(gw) 53 | visualizeTN(gw) 54 | 55 | k,n = symbols('k,n') 56 | Eg = integrateHaarUnitary(gw, ["U", [n, k], [n, k], n*k]) 57 | print(Eg) 58 | visualizeTN(Eg) 59 | 60 | 61 | ##### 6.3 ##### 62 | from IHU_source import * 63 | 64 | e1 = [["X", 1, "out", 1], ["U", 1, "in", 1]] 65 | e2 = [["U*", 1, "out", 1], ["X", 1, "in", 1]] 66 | e3 = [["X", 1, "out", 2], ["U", 2, "in", 1]] 67 | e4 = [["U*", 2, "out", 1], ["X", 1, "in", 2]] 68 | g = [e1, e2, e3, e4] 69 | gw = [g,1] 70 | print(gw) 71 | visualizeTN(gw) 72 | 73 | d = symbols('d') 74 | Eg = integrateHaarUnitary(gw, ["U", [d], [d], d]) 75 | print(Eg) 76 | visualizeTN(Eg) 77 | 78 | 79 | #### 6.4 #### 80 | from IHU_source import * 81 | d,k,n = symbols('d,k,n') 82 | e1 = [["U*", 2, "out", 1], ["U", 1, "in", 1]] 83 | e2 = [["U*", 1, "out", 1], ["U", 2, "in", 1]] 84 | e3 = [["U", 1, "out", 1], ["U*", 1, "in", 1]] 85 | e4 = [["U", 2, "out", 1], ["U*", 2, "in", 1]] 86 | e5 = [["U", 1, "out", 2], ["U*", 2, "in", 2]] 87 | e6 = [["U", 2, "out", 2], ["U*", 1, "in", 2]] 88 | g = [e1, e2, e3, e4, e5, e6] 89 | gw = [[g, 1/(d* k)]] 90 | print(gw) 91 | visualizeTN(gw) 92 | 93 | Eg = integrateHaarUnitary(gw, ["U", [d], [n, k], n*k]) 94 | print(Eg) 95 | #visualizeTN(Eg) 96 | 97 | from sympy import limit, oo 98 | t = symbols('t') 99 | Egt = Eg[0][1].subs(d,t*k*n) 100 | Egt_lim = simplify(limit(Egt,n,oo)) 101 | print(Egt_lim) 102 | 103 | -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions7.html: -------------------------------------------------------------------------------- 1 |

Weingarten functions for p = 7

Below are the values of the Weingarten function for permutation size p = 7. The input is given as a partition of p. You can also download them as a text file or as a python pickle file. 2 | 3 | \[\operatorname{Wg}([1, 1, 1, 1, 1, 1, 1]) = \frac{n^{8} - 71 n^{6} + 1568 n^{4} - 11398 n^{2} + 15780}{n \left(n^{14} - 92 n^{12} + 3094 n^{10} - 47476 n^{8} + 340769 n^{6} - 1069432 n^{4} + 1291536 n^{2} - 518400\right)}\] 4 | \[\operatorname{Wg}([2, 1, 1, 1, 1, 1]) = \frac{- n^{8} + 61 n^{6} - 1058 n^{4} + 4958 n^{2} - 1440}{n^{2} \left(n^{14} - 92 n^{12} + 3094 n^{10} - 47476 n^{8} + 340769 n^{6} - 1069432 n^{4} + 1291536 n^{2} - 518400\right)}\] 5 | \[\operatorname{Wg}([3, 1, 1, 1, 1]) = \frac{2 n^{6} - 102 n^{4} + 1288 n^{2} - 2868}{n \left(n^{14} - 92 n^{12} + 3094 n^{10} - 47476 n^{8} + 340769 n^{6} - 1069432 n^{4} + 1291536 n^{2} - 518400\right)}\] 6 | \[\operatorname{Wg}([2, 2, 1, 1, 1]) = \frac{n^{6} - 45 n^{4} + 488 n^{2} - 1284}{n \left(n^{14} - 92 n^{12} + 3094 n^{10} - 47476 n^{8} + 340769 n^{6} - 1069432 n^{4} + 1291536 n^{2} - 518400\right)}\] 7 | \[\operatorname{Wg}([4, 1, 1, 1]) = \frac{- 5 n^{6} + 207 n^{4} - 1762 n^{2} + 720}{n^{2} \left(n^{14} - 92 n^{12} + 3094 n^{10} - 47476 n^{8} + 340769 n^{6} - 1069432 n^{4} + 1291536 n^{2} - 518400\right)}\] 8 | \[\operatorname{Wg}([3, 2, 1, 1]) = - \frac{2}{n^{10} - 66 n^{8} + 1353 n^{6} - 10648 n^{4} + 30096 n^{2} - 20736}\] 9 | \[\operatorname{Wg}([5, 1, 1]) = \frac{14 n^{2} - 84}{n \left(n^{12} - 67 n^{10} + 1419 n^{8} - 12001 n^{6} + 40744 n^{4} - 50832 n^{2} + 20736\right)}\] 10 | \[\operatorname{Wg}([2, 2, 2, 1]) = \frac{- n^{6} + 23 n^{4} - 382 n^{2} + 2880}{n^{2} \left(n^{14} - 92 n^{12} + 3094 n^{10} - 47476 n^{8} + 340769 n^{6} - 1069432 n^{4} + 1291536 n^{2} - 518400\right)}\] 11 | \[\operatorname{Wg}([4, 2, 1]) = \frac{5 n^{2} + 51}{n \left(n^{12} - 76 n^{10} + 1878 n^{8} - 17428 n^{6} + 61921 n^{4} - 78696 n^{2} + 32400\right)}\] 12 | \[\operatorname{Wg}([3, 3, 1]) = \frac{4 n^{4} + 32 n^{2} - 1716}{n \left(n^{14} - 92 n^{12} + 3094 n^{10} - 47476 n^{8} + 340769 n^{6} - 1069432 n^{4} + 1291536 n^{2} - 518400\right)}\] 13 | \[\operatorname{Wg}([6, 1]) = \frac{- 42 n^{2} + 720}{n^{2} \left(n^{12} - 91 n^{10} + 3003 n^{8} - 44473 n^{6} + 296296 n^{4} - 773136 n^{2} + 518400\right)}\] 14 | \[\operatorname{Wg}([3, 2, 2]) = \frac{2 n^{4} + 10 n^{2} + 1668}{n \left(n^{14} - 92 n^{12} + 3094 n^{10} - 47476 n^{8} + 340769 n^{6} - 1069432 n^{4} + 1291536 n^{2} - 518400\right)}\] 15 | \[\operatorname{Wg}([5, 2]) = \frac{- 14 n^{4} - 226 n^{2} - 1440}{n^{2} \left(n^{14} - 92 n^{12} + 3094 n^{10} - 47476 n^{8} + 340769 n^{6} - 1069432 n^{4} + 1291536 n^{2} - 518400\right)}\] 16 | \[\operatorname{Wg}([4, 3]) = \frac{- 10 n^{4} - 470 n^{2} + 2160}{n^{2} \left(n^{14} - 92 n^{12} + 3094 n^{10} - 47476 n^{8} + 340769 n^{6} - 1069432 n^{4} + 1291536 n^{2} - 518400\right)}\] 17 | \[\operatorname{Wg}([7]) = \frac{132}{n \left(n^{12} - 91 n^{10} + 3003 n^{8} - 44473 n^{6} + 296296 n^{4} - 773136 n^{2} + 518400\right)}\] 18 | -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions8.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1, 1, 1, 1, 1, 1, 1], (n**12 - 117*n**10 + 4792*n**8 - 82644*n**6 + 573772*n**4 - 1337484*n**2 + 771120)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[2, 1, 1, 1, 1, 1, 1], (-n**10 + 105*n**8 - 3682*n**6 + 50490*n**4 - 247552*n**2 + 331680)/(n*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[3, 1, 1, 1, 1, 1], (2*n**10 - 185*n**8 + 5359*n**6 - 54370*n**4 + 167634*n**2 - 128520)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[2, 2, 1, 1, 1, 1], (n**10 - 87*n**8 + 2356*n**6 - 23058*n**4 + 76228*n**2 - 5040)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[4, 1, 1, 1, 1], (-5*n**8 + 401*n**6 - 9322*n**4 + 65494*n**2 - 106968)/(n*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[3, 2, 1, 1, 1], (-2*n**8 + 131*n**6 - 2197*n**4 + 10084*n**2 - 28176)/(n*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[5, 1, 1, 1], 14*(n**8 - 67*n**6 + 1184*n**4 - 5438*n**2 + 4680)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[2, 2, 2, 1, 1], (-n**4 + 23*n**2 - 310)/(n*(n**14 - 105*n**12 + 3822*n**10 - 61490*n**8 + 453453*n**6 - 1442805*n**4 + 1752724*n**2 - 705600))], [[4, 2, 1, 1], (5*n**4 - 13*n**2 - 280)/(n**2*(n**14 - 105*n**12 + 3822*n**10 - 61490*n**8 + 453453*n**6 - 1442805*n**4 + 1752724*n**2 - 705600))], [[3, 3, 1, 1], 2*(2*n**6 + 14*n**4 - 403*n**2 + 315)/(n**2*(n**16 - 109*n**14 + 4242*n**12 - 76778*n**10 + 699413*n**8 - 3256617*n**6 + 7523944*n**4 - 7716496*n**2 + 2822400))], [[6, 1, 1], (-42*n**2 + 474)/(n*(n**14 - 105*n**12 + 3822*n**10 - 61490*n**8 + 453453*n**6 - 1442805*n**4 + 1752724*n**2 - 705600))], [[3, 2, 2, 1], (2*n**8 - 65*n**6 + 1753*n**4 - 34450*n**2 + 83160)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[5, 2, 1], -(14*n**4 + 100*n**2 + 96)/(n*(n**16 - 120*n**14 + 5166*n**12 - 100340*n**10 + 954921*n**8 - 4562460*n**6 + 10700656*n**4 - 11062080*n**2 + 4064256))], [[4, 3, 1], (-10*n**6 - 127*n**4 + 12605*n**2 - 62868)/(n*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[7, 1], 33*(4*n**2 - 105)/(n**2*(n**14 - 140*n**12 + 7462*n**10 - 191620*n**8 + 2475473*n**6 - 15291640*n**4 + 38402064*n**2 - 25401600))], [[2, 2, 2, 2], (n**8 - 33*n**6 + 1404*n**4 + 23828*n**2 + 65520)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[4, 2, 2], -(5*n**6 + 5*n**4 + 6590*n**2 + 3480)/(n*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[3, 3, 2], (-4*n**6 - 52*n**4 - 8056*n**2 + 63552)/(n*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[6, 2], 42*(n**4 + 14*n**2 + 345)/(n**2*(n**16 - 141*n**14 + 7602*n**12 - 199082*n**10 + 2667093*n**8 - 17767113*n**6 + 53693704*n**4 - 63803664*n**2 + 25401600))], [[5, 3], 7*(4*n**6 + 235*n**4 - 2219*n**2 + 540)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[4, 4], 5*(5*n**6 + 395*n**4 - 5440*n**2 + 27216)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[8], -429/(n*(n**14 - 140*n**12 + 7462*n**10 - 191620*n**8 + 2475473*n**6 - 15291640*n**4 + 38402064*n**2 - 25401600))]] -------------------------------------------------------------------------------- /MATHEMATICA/precomputedWG/functions8.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1, 1, 1, 1, 1, 1, 1], (n**12 - 117*n**10 + 4792*n**8 - 82644*n**6 + 573772*n**4 - 1337484*n**2 + 771120)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[2, 1, 1, 1, 1, 1, 1], (-n**10 + 105*n**8 - 3682*n**6 + 50490*n**4 - 247552*n**2 + 331680)/(n*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[3, 1, 1, 1, 1, 1], (2*n**10 - 185*n**8 + 5359*n**6 - 54370*n**4 + 167634*n**2 - 128520)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[2, 2, 1, 1, 1, 1], (n**10 - 87*n**8 + 2356*n**6 - 23058*n**4 + 76228*n**2 - 5040)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[4, 1, 1, 1, 1], (-5*n**8 + 401*n**6 - 9322*n**4 + 65494*n**2 - 106968)/(n*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[3, 2, 1, 1, 1], (-2*n**8 + 131*n**6 - 2197*n**4 + 10084*n**2 - 28176)/(n*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[5, 1, 1, 1], 14*(n**8 - 67*n**6 + 1184*n**4 - 5438*n**2 + 4680)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[2, 2, 2, 1, 1], (-n**4 + 23*n**2 - 310)/(n*(n**14 - 105*n**12 + 3822*n**10 - 61490*n**8 + 453453*n**6 - 1442805*n**4 + 1752724*n**2 - 705600))], [[4, 2, 1, 1], (5*n**4 - 13*n**2 - 280)/(n**2*(n**14 - 105*n**12 + 3822*n**10 - 61490*n**8 + 453453*n**6 - 1442805*n**4 + 1752724*n**2 - 705600))], [[3, 3, 1, 1], 2*(2*n**6 + 14*n**4 - 403*n**2 + 315)/(n**2*(n**16 - 109*n**14 + 4242*n**12 - 76778*n**10 + 699413*n**8 - 3256617*n**6 + 7523944*n**4 - 7716496*n**2 + 2822400))], [[6, 1, 1], (-42*n**2 + 474)/(n*(n**14 - 105*n**12 + 3822*n**10 - 61490*n**8 + 453453*n**6 - 1442805*n**4 + 1752724*n**2 - 705600))], [[3, 2, 2, 1], (2*n**8 - 65*n**6 + 1753*n**4 - 34450*n**2 + 83160)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[5, 2, 1], -(14*n**4 + 100*n**2 + 96)/(n*(n**16 - 120*n**14 + 5166*n**12 - 100340*n**10 + 954921*n**8 - 4562460*n**6 + 10700656*n**4 - 11062080*n**2 + 4064256))], [[4, 3, 1], (-10*n**6 - 127*n**4 + 12605*n**2 - 62868)/(n*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[7, 1], 33*(4*n**2 - 105)/(n**2*(n**14 - 140*n**12 + 7462*n**10 - 191620*n**8 + 2475473*n**6 - 15291640*n**4 + 38402064*n**2 - 25401600))], [[2, 2, 2, 2], (n**8 - 33*n**6 + 1404*n**4 + 23828*n**2 + 65520)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[4, 2, 2], -(5*n**6 + 5*n**4 + 6590*n**2 + 3480)/(n*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[3, 3, 2], (-4*n**6 - 52*n**4 - 8056*n**2 + 63552)/(n*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[6, 2], 42*(n**4 + 14*n**2 + 345)/(n**2*(n**16 - 141*n**14 + 7602*n**12 - 199082*n**10 + 2667093*n**8 - 17767113*n**6 + 53693704*n**4 - 63803664*n**2 + 25401600))], [[5, 3], 7*(4*n**6 + 235*n**4 - 2219*n**2 + 540)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[4, 4], 5*(5*n**6 + 395*n**4 - 5440*n**2 + 27216)/(n**2*(n**18 - 145*n**16 + 8166*n**14 - 229490*n**12 + 3463421*n**10 - 28435485*n**8 + 124762156*n**6 - 278578480*n**4 + 280616256*n**2 - 101606400))], [[8], -429/(n*(n**14 - 140*n**12 + 7462*n**10 - 191620*n**8 + 2475473*n**6 - 15291640*n**4 + 38402064*n**2 - 25401600))]] -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions8.html: -------------------------------------------------------------------------------- 1 |

Weingarten functions for p = 8

Below are the values of the Weingarten function for permutation size p = 8. The input is given as a partition of p. You can also download them as a text file or as a python pickle file. 2 | 3 | \[\operatorname{Wg}([1, 1, 1, 1, 1, 1, 1, 1]) = \frac{n^{12} - 117 n^{10} + 4792 n^{8} - 82644 n^{6} + 573772 n^{4} - 1337484 n^{2} + 771120}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\] 4 | \[\operatorname{Wg}([2, 1, 1, 1, 1, 1, 1]) = \frac{- n^{10} + 105 n^{8} - 3682 n^{6} + 50490 n^{4} - 247552 n^{2} + 331680}{n \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\] 5 | \[\operatorname{Wg}([3, 1, 1, 1, 1, 1]) = \frac{2 n^{10} - 185 n^{8} + 5359 n^{6} - 54370 n^{4} + 167634 n^{2} - 128520}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\] 6 | \[\operatorname{Wg}([2, 2, 1, 1, 1, 1]) = \frac{n^{10} - 87 n^{8} + 2356 n^{6} - 23058 n^{4} + 76228 n^{2} - 5040}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\] 7 | \[\operatorname{Wg}([4, 1, 1, 1, 1]) = \frac{- 5 n^{8} + 401 n^{6} - 9322 n^{4} + 65494 n^{2} - 106968}{n \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\] 8 | \[\operatorname{Wg}([3, 2, 1, 1, 1]) = \frac{- 2 n^{8} + 131 n^{6} - 2197 n^{4} + 10084 n^{2} - 28176}{n \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\] 9 | \[\operatorname{Wg}([5, 1, 1, 1]) = \frac{14 n^{8} - 938 n^{6} + 16576 n^{4} - 76132 n^{2} + 65520}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\] 10 | \[\operatorname{Wg}([2, 2, 2, 1, 1]) = \frac{- n^{4} + 23 n^{2} - 310}{n \left(n^{14} - 105 n^{12} + 3822 n^{10} - 61490 n^{8} + 453453 n^{6} - 1442805 n^{4} + 1752724 n^{2} - 705600\right)}\] 11 | \[\operatorname{Wg}([4, 2, 1, 1]) = \frac{5 n^{4} - 13 n^{2} - 280}{n^{2} \left(n^{14} - 105 n^{12} + 3822 n^{10} - 61490 n^{8} + 453453 n^{6} - 1442805 n^{4} + 1752724 n^{2} - 705600\right)}\] 12 | \[\operatorname{Wg}([3, 3, 1, 1]) = \frac{4 n^{6} + 28 n^{4} - 806 n^{2} + 630}{n^{2} \left(n^{16} - 109 n^{14} + 4242 n^{12} - 76778 n^{10} + 699413 n^{8} - 3256617 n^{6} + 7523944 n^{4} - 7716496 n^{2} + 2822400\right)}\] 13 | \[\operatorname{Wg}([6, 1, 1]) = \frac{- 42 n^{2} + 474}{n \left(n^{14} - 105 n^{12} + 3822 n^{10} - 61490 n^{8} + 453453 n^{6} - 1442805 n^{4} + 1752724 n^{2} - 705600\right)}\] 14 | \[\operatorname{Wg}([3, 2, 2, 1]) = \frac{2 n^{8} - 65 n^{6} + 1753 n^{4} - 34450 n^{2} + 83160}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\] 15 | \[\operatorname{Wg}([5, 2, 1]) = \frac{- 14 n^{4} - 100 n^{2} - 96}{n \left(n^{16} - 120 n^{14} + 5166 n^{12} - 100340 n^{10} + 954921 n^{8} - 4562460 n^{6} + 10700656 n^{4} - 11062080 n^{2} + 4064256\right)}\] 16 | \[\operatorname{Wg}([4, 3, 1]) = \frac{- 10 n^{6} - 127 n^{4} + 12605 n^{2} - 62868}{n \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\] 17 | \[\operatorname{Wg}([7, 1]) = \frac{132 n^{2} - 3465}{n^{2} \left(n^{14} - 140 n^{12} + 7462 n^{10} - 191620 n^{8} + 2475473 n^{6} - 15291640 n^{4} + 38402064 n^{2} - 25401600\right)}\] 18 | \[\operatorname{Wg}([2, 2, 2, 2]) = \frac{n^{8} - 33 n^{6} + 1404 n^{4} + 23828 n^{2} + 65520}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\] 19 | \[\operatorname{Wg}([4, 2, 2]) = \frac{- 5 n^{6} - 5 n^{4} - 6590 n^{2} - 3480}{n \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\] 20 | \[\operatorname{Wg}([3, 3, 2]) = \frac{- 4 n^{6} - 52 n^{4} - 8056 n^{2} + 63552}{n \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\] 21 | \[\operatorname{Wg}([6, 2]) = \frac{42 n^{4} + 588 n^{2} + 14490}{n^{2} \left(n^{16} - 141 n^{14} + 7602 n^{12} - 199082 n^{10} + 2667093 n^{8} - 17767113 n^{6} + 53693704 n^{4} - 63803664 n^{2} + 25401600\right)}\] 22 | \[\operatorname{Wg}([5, 3]) = \frac{28 n^{6} + 1645 n^{4} - 15533 n^{2} + 3780}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\] 23 | \[\operatorname{Wg}([4, 4]) = \frac{25 n^{6} + 1975 n^{4} - 27200 n^{2} + 136080}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\] 24 | \[\operatorname{Wg}([8]) = - \frac{429}{n \left(n^{14} - 140 n^{12} + 7462 n^{10} - 191620 n^{8} + 2475473 n^{6} - 15291640 n^{4} + 38402064 n^{2} - 25401600\right)}\] 25 | -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions9.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1, 1, 1, 1, 1, 1, 1, 1], (n**14 - 173*n**12 + 11008*n**10 - 317396*n**8 + 4114204*n**6 - 20599916*n**4 + 29866032*n**2 - 7902720)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[2, 1, 1, 1, 1, 1, 1, 1], (-n**10 + 155*n**8 - 8372*n**6 + 185110*n**4 - 1441932*n**2 + 1295280)/(n**2*(n**18 - 205*n**16 + 16626*n**14 - 685610*n**12 + 15408341*n**10 - 188461065*n**8 + 1190788936*n**6 - 3500200720*n**4 + 4108836096*n**2 - 1625702400))], [[3, 1, 1, 1, 1, 1, 1], 2*(n**12 - 144*n**10 + 7057*n**8 - 137988*n**6 + 966922*n**4 - 1763208*n**2 + 564480)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[2, 2, 1, 1, 1, 1, 1], (n**12 - 139*n**10 + 6552*n**8 - 124098*n**6 + 874052*n**4 - 1498928*n**2 - 376320)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[4, 1, 1, 1, 1, 1], (-5*n**10 + 645*n**8 - 26870*n**6 + 405150*n**4 - 1766600*n**2 + 1538880)/(n**2*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[3, 2, 1, 1, 1, 1], (-2*n**6 + 150*n**4 - 1864*n**2 + 3444)/(n**2*(n**16 - 169*n**14 + 10542*n**12 - 306098*n**10 + 4388813*n**8 - 30463797*n**6 + 94092244*n**4 - 112879936*n**2 + 45158400))], [[5, 1, 1, 1, 1], 14*(n**10 - 113*n**8 + 3870*n**6 - 42502*n**4 + 100184*n**2 - 26880)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[2, 2, 2, 1, 1, 1], (-n**10 + 113*n**8 - 4284*n**6 + 69868*n**4 - 440000*n**2 - 321216)/(n**2*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[4, 2, 1, 1, 1], (5*n**8 - 427*n**6 + 8350*n**4 - 4568*n**2 - 94080)/(n**3*(n**18 - 205*n**16 + 16626*n**14 - 685610*n**12 + 15408341*n**10 - 188461065*n**8 + 1190788936*n**6 - 3500200720*n**4 + 4108836096*n**2 - 1625702400))], [[3, 3, 1, 1, 1], 2*(2*n**6 - 144*n**4 + 1153*n**2 + 36789)/(n*(n**18 - 205*n**16 + 16626*n**14 - 685610*n**12 + 15408341*n**10 - 188461065*n**8 + 1190788936*n**6 - 3500200720*n**4 + 4108836096*n**2 - 1625702400))], [[6, 1, 1, 1], (-42*n**8 + 3966*n**6 - 103620*n**4 + 735744*n**2 - 923328)/(n**2*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[3, 2, 2, 1, 1], 2*(n**8 - 31*n**6 + 814*n**4 - 7144*n**2 + 3840)/(n**3*(n**18 - 160*n**16 + 9606*n**14 - 281420*n**12 + 4361201*n**10 - 36395580*n**8 + 161249776*n**6 - 362117440*n**4 + 365884416*n**2 - 132710400))], [[5, 2, 1, 1], (-14*n**4 + 124*n**2 + 1360)/(n**2*(n**16 - 156*n**14 + 8982*n**12 - 245492*n**10 + 3379233*n**8 - 22878648*n**6 + 69735184*n**4 - 83176704*n**2 + 33177600))], [[4, 3, 1, 1], (-10*n**6 - 174*n**4 + 7240*n**2 - 12096)/(n**2*(n**18 - 160*n**16 + 9606*n**14 - 281420*n**12 + 4361201*n**10 - 36395580*n**8 + 161249776*n**6 - 362117440*n**4 + 365884416*n**2 - 132710400))], [[7, 1, 1], 66*(2*n**2 - 37)/(n*(n**16 - 156*n**14 + 8982*n**12 - 245492*n**10 + 3379233*n**8 - 22878648*n**6 + 69735184*n**4 - 83176704*n**2 + 33177600))], [[2, 2, 2, 2, 1], (n**10 - 81*n**8 + 2892*n**6 - 22924*n**4 - 534288*n**2 + 1128960)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[4, 2, 2, 1], (-5*n**8 + 219*n**6 - 6926*n**4 + 181656*n**2 - 23744)/(n**2*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[3, 3, 2, 1], -(4*n**4 + 32*n**2 + 7308)/(n**2*(n**16 - 169*n**14 + 10542*n**12 - 306098*n**10 + 4388813*n**8 - 30463797*n**6 + 94092244*n**4 - 112879936*n**2 + 45158400))], [[6, 2, 1], 6*(7*n**4 + 34*n**2 + 1327)/(n*(n**18 - 173*n**16 + 11218*n**14 - 348266*n**12 + 5613205*n**10 - 48019049*n**8 + 215947432*n**6 - 489248912*n**4 + 496678144*n**2 - 180633600))], [[5, 3, 1], 7*(4*n**8 + 69*n**6 - 9549*n**4 + 51716*n**2 + 26880)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[4, 4, 1], 5*(5*n**8 + 187*n**6 - 17840*n**4 + 147344*n**2 - 75264)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[8, 1], (-429*n**2 + 16016)/(n**2*(n**16 - 204*n**14 + 16422*n**12 - 669188*n**10 + 14739153*n**8 - 173721912*n**6 + 1017067024*n**4 - 2483133696*n**2 + 1625702400))], [[3, 2, 2, 2], (-2*n**8 + 78*n**6 - 5388*n**4 - 115648*n**2 - 665280)/(n**2*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[5, 2, 2], 2*(7*n**8 - 33*n**6 + 13458*n**4 + 144488*n**2 - 188160)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[4, 3, 2], 10*(n**6 + 27*n**4 + 3332*n**2 - 9408)/(n**3*(n**18 - 205*n**16 + 16626*n**14 - 685610*n**12 + 15408341*n**10 - 188461065*n**8 + 1190788936*n**6 - 3500200720*n**4 + 4108836096*n**2 - 1625702400))], [[7, 2], -(132*n**4 + 1100*n**2 + 109648)/(n**2*(n**18 - 205*n**16 + 16626*n**14 - 685610*n**12 + 15408341*n**10 - 188461065*n**8 + 1190788936*n**6 - 3500200720*n**4 + 4108836096*n**2 - 1625702400))], [[3, 3, 3], 8*(n**8 + 31*n**6 + 6124*n**4 - 103176*n**2 - 141120)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[6, 3], (-84*n**6 - 5964*n**4 + 41664*n**2 + 538944)/(n**2*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[5, 4], (-70*n**6 - 8050*n**4 + 142520*n**2 - 1041600)/(n**2*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[9], 1430/(n*(n**16 - 204*n**14 + 16422*n**12 - 669188*n**10 + 14739153*n**8 - 173721912*n**6 + 1017067024*n**4 - 2483133696*n**2 + 1625702400))]] -------------------------------------------------------------------------------- /MATHEMATICA/precomputedWG/functions9.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1, 1, 1, 1, 1, 1, 1, 1], (n**14 - 173*n**12 + 11008*n**10 - 317396*n**8 + 4114204*n**6 - 20599916*n**4 + 29866032*n**2 - 7902720)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[2, 1, 1, 1, 1, 1, 1, 1], (-n**10 + 155*n**8 - 8372*n**6 + 185110*n**4 - 1441932*n**2 + 1295280)/(n**2*(n**18 - 205*n**16 + 16626*n**14 - 685610*n**12 + 15408341*n**10 - 188461065*n**8 + 1190788936*n**6 - 3500200720*n**4 + 4108836096*n**2 - 1625702400))], [[3, 1, 1, 1, 1, 1, 1], 2*(n**12 - 144*n**10 + 7057*n**8 - 137988*n**6 + 966922*n**4 - 1763208*n**2 + 564480)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[2, 2, 1, 1, 1, 1, 1], (n**12 - 139*n**10 + 6552*n**8 - 124098*n**6 + 874052*n**4 - 1498928*n**2 - 376320)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[4, 1, 1, 1, 1, 1], (-5*n**10 + 645*n**8 - 26870*n**6 + 405150*n**4 - 1766600*n**2 + 1538880)/(n**2*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[3, 2, 1, 1, 1, 1], (-2*n**6 + 150*n**4 - 1864*n**2 + 3444)/(n**2*(n**16 - 169*n**14 + 10542*n**12 - 306098*n**10 + 4388813*n**8 - 30463797*n**6 + 94092244*n**4 - 112879936*n**2 + 45158400))], [[5, 1, 1, 1, 1], 14*(n**10 - 113*n**8 + 3870*n**6 - 42502*n**4 + 100184*n**2 - 26880)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[2, 2, 2, 1, 1, 1], (-n**10 + 113*n**8 - 4284*n**6 + 69868*n**4 - 440000*n**2 - 321216)/(n**2*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[4, 2, 1, 1, 1], (5*n**8 - 427*n**6 + 8350*n**4 - 4568*n**2 - 94080)/(n**3*(n**18 - 205*n**16 + 16626*n**14 - 685610*n**12 + 15408341*n**10 - 188461065*n**8 + 1190788936*n**6 - 3500200720*n**4 + 4108836096*n**2 - 1625702400))], [[3, 3, 1, 1, 1], 2*(2*n**6 - 144*n**4 + 1153*n**2 + 36789)/(n*(n**18 - 205*n**16 + 16626*n**14 - 685610*n**12 + 15408341*n**10 - 188461065*n**8 + 1190788936*n**6 - 3500200720*n**4 + 4108836096*n**2 - 1625702400))], [[6, 1, 1, 1], (-42*n**8 + 3966*n**6 - 103620*n**4 + 735744*n**2 - 923328)/(n**2*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[3, 2, 2, 1, 1], 2*(n**8 - 31*n**6 + 814*n**4 - 7144*n**2 + 3840)/(n**3*(n**18 - 160*n**16 + 9606*n**14 - 281420*n**12 + 4361201*n**10 - 36395580*n**8 + 161249776*n**6 - 362117440*n**4 + 365884416*n**2 - 132710400))], [[5, 2, 1, 1], (-14*n**4 + 124*n**2 + 1360)/(n**2*(n**16 - 156*n**14 + 8982*n**12 - 245492*n**10 + 3379233*n**8 - 22878648*n**6 + 69735184*n**4 - 83176704*n**2 + 33177600))], [[4, 3, 1, 1], (-10*n**6 - 174*n**4 + 7240*n**2 - 12096)/(n**2*(n**18 - 160*n**16 + 9606*n**14 - 281420*n**12 + 4361201*n**10 - 36395580*n**8 + 161249776*n**6 - 362117440*n**4 + 365884416*n**2 - 132710400))], [[7, 1, 1], 66*(2*n**2 - 37)/(n*(n**16 - 156*n**14 + 8982*n**12 - 245492*n**10 + 3379233*n**8 - 22878648*n**6 + 69735184*n**4 - 83176704*n**2 + 33177600))], [[2, 2, 2, 2, 1], (n**10 - 81*n**8 + 2892*n**6 - 22924*n**4 - 534288*n**2 + 1128960)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[4, 2, 2, 1], (-5*n**8 + 219*n**6 - 6926*n**4 + 181656*n**2 - 23744)/(n**2*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[3, 3, 2, 1], -(4*n**4 + 32*n**2 + 7308)/(n**2*(n**16 - 169*n**14 + 10542*n**12 - 306098*n**10 + 4388813*n**8 - 30463797*n**6 + 94092244*n**4 - 112879936*n**2 + 45158400))], [[6, 2, 1], 6*(7*n**4 + 34*n**2 + 1327)/(n*(n**18 - 173*n**16 + 11218*n**14 - 348266*n**12 + 5613205*n**10 - 48019049*n**8 + 215947432*n**6 - 489248912*n**4 + 496678144*n**2 - 180633600))], [[5, 3, 1], 7*(4*n**8 + 69*n**6 - 9549*n**4 + 51716*n**2 + 26880)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[4, 4, 1], 5*(5*n**8 + 187*n**6 - 17840*n**4 + 147344*n**2 - 75264)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[8, 1], (-429*n**2 + 16016)/(n**2*(n**16 - 204*n**14 + 16422*n**12 - 669188*n**10 + 14739153*n**8 - 173721912*n**6 + 1017067024*n**4 - 2483133696*n**2 + 1625702400))], [[3, 2, 2, 2], (-2*n**8 + 78*n**6 - 5388*n**4 - 115648*n**2 - 665280)/(n**2*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[5, 2, 2], 2*(7*n**8 - 33*n**6 + 13458*n**4 + 144488*n**2 - 188160)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[4, 3, 2], 10*(n**6 + 27*n**4 + 3332*n**2 - 9408)/(n**3*(n**18 - 205*n**16 + 16626*n**14 - 685610*n**12 + 15408341*n**10 - 188461065*n**8 + 1190788936*n**6 - 3500200720*n**4 + 4108836096*n**2 - 1625702400))], [[7, 2], -(132*n**4 + 1100*n**2 + 109648)/(n**2*(n**18 - 205*n**16 + 16626*n**14 - 685610*n**12 + 15408341*n**10 - 188461065*n**8 + 1190788936*n**6 - 3500200720*n**4 + 4108836096*n**2 - 1625702400))], [[3, 3, 3], 8*(n**8 + 31*n**6 + 6124*n**4 - 103176*n**2 - 141120)/(n**3*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[6, 3], (-84*n**6 - 5964*n**4 + 41664*n**2 + 538944)/(n**2*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[5, 4], (-70*n**6 - 8050*n**4 + 142520*n**2 - 1041600)/(n**2*(n**20 - 209*n**18 + 17446*n**16 - 752114*n**14 + 18150781*n**12 - 250094429*n**10 + 1944633196*n**8 - 8263356464*n**6 + 18109638976*n**4 - 18061046784*n**2 + 6502809600))], [[9], 1430/(n*(n**16 - 204*n**14 + 16422*n**12 - 669188*n**10 + 14739153*n**8 - 173721912*n**6 + 1017067024*n**4 - 2483133696*n**2 + 1625702400))]] -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions9.html: -------------------------------------------------------------------------------- 1 |

Weingarten functions for p = 9

Below are the values of the Weingarten function for permutation size p = 9. The input is given as a partition of p. You can also download them as a text file or as a python pickle file. 2 | 3 | \[\operatorname{Wg}([1, 1, 1, 1, 1, 1, 1, 1, 1]) = \frac{n^{14} - 173 n^{12} + 11008 n^{10} - 317396 n^{8} + 4114204 n^{6} - 20599916 n^{4} + 29866032 n^{2} - 7902720}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 4 | \[\operatorname{Wg}([2, 1, 1, 1, 1, 1, 1, 1]) = \frac{- n^{10} + 155 n^{8} - 8372 n^{6} + 185110 n^{4} - 1441932 n^{2} + 1295280}{n^{2} \left(n^{18} - 205 n^{16} + 16626 n^{14} - 685610 n^{12} + 15408341 n^{10} - 188461065 n^{8} + 1190788936 n^{6} - 3500200720 n^{4} + 4108836096 n^{2} - 1625702400\right)}\] 5 | \[\operatorname{Wg}([3, 1, 1, 1, 1, 1, 1]) = \frac{2 n^{12} - 288 n^{10} + 14114 n^{8} - 275976 n^{6} + 1933844 n^{4} - 3526416 n^{2} + 1128960}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 6 | \[\operatorname{Wg}([2, 2, 1, 1, 1, 1, 1]) = \frac{n^{12} - 139 n^{10} + 6552 n^{8} - 124098 n^{6} + 874052 n^{4} - 1498928 n^{2} - 376320}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 7 | \[\operatorname{Wg}([4, 1, 1, 1, 1, 1]) = \frac{- 5 n^{10} + 645 n^{8} - 26870 n^{6} + 405150 n^{4} - 1766600 n^{2} + 1538880}{n^{2} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 8 | \[\operatorname{Wg}([3, 2, 1, 1, 1, 1]) = \frac{- 2 n^{6} + 150 n^{4} - 1864 n^{2} + 3444}{n^{2} \left(n^{16} - 169 n^{14} + 10542 n^{12} - 306098 n^{10} + 4388813 n^{8} - 30463797 n^{6} + 94092244 n^{4} - 112879936 n^{2} + 45158400\right)}\] 9 | \[\operatorname{Wg}([5, 1, 1, 1, 1]) = \frac{14 n^{10} - 1582 n^{8} + 54180 n^{6} - 595028 n^{4} + 1402576 n^{2} - 376320}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 10 | \[\operatorname{Wg}([2, 2, 2, 1, 1, 1]) = \frac{- n^{10} + 113 n^{8} - 4284 n^{6} + 69868 n^{4} - 440000 n^{2} - 321216}{n^{2} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 11 | \[\operatorname{Wg}([4, 2, 1, 1, 1]) = \frac{5 n^{8} - 427 n^{6} + 8350 n^{4} - 4568 n^{2} - 94080}{n^{3} \left(n^{18} - 205 n^{16} + 16626 n^{14} - 685610 n^{12} + 15408341 n^{10} - 188461065 n^{8} + 1190788936 n^{6} - 3500200720 n^{4} + 4108836096 n^{2} - 1625702400\right)}\] 12 | \[\operatorname{Wg}([3, 3, 1, 1, 1]) = \frac{4 n^{6} - 288 n^{4} + 2306 n^{2} + 73578}{n \left(n^{18} - 205 n^{16} + 16626 n^{14} - 685610 n^{12} + 15408341 n^{10} - 188461065 n^{8} + 1190788936 n^{6} - 3500200720 n^{4} + 4108836096 n^{2} - 1625702400\right)}\] 13 | \[\operatorname{Wg}([6, 1, 1, 1]) = \frac{- 42 n^{8} + 3966 n^{6} - 103620 n^{4} + 735744 n^{2} - 923328}{n^{2} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 14 | \[\operatorname{Wg}([3, 2, 2, 1, 1]) = \frac{2 n^{8} - 62 n^{6} + 1628 n^{4} - 14288 n^{2} + 7680}{n^{3} \left(n^{18} - 160 n^{16} + 9606 n^{14} - 281420 n^{12} + 4361201 n^{10} - 36395580 n^{8} + 161249776 n^{6} - 362117440 n^{4} + 365884416 n^{2} - 132710400\right)}\] 15 | \[\operatorname{Wg}([5, 2, 1, 1]) = \frac{- 14 n^{4} + 124 n^{2} + 1360}{n^{2} \left(n^{16} - 156 n^{14} + 8982 n^{12} - 245492 n^{10} + 3379233 n^{8} - 22878648 n^{6} + 69735184 n^{4} - 83176704 n^{2} + 33177600\right)}\] 16 | \[\operatorname{Wg}([4, 3, 1, 1]) = \frac{- 10 n^{6} - 174 n^{4} + 7240 n^{2} - 12096}{n^{2} \left(n^{18} - 160 n^{16} + 9606 n^{14} - 281420 n^{12} + 4361201 n^{10} - 36395580 n^{8} + 161249776 n^{6} - 362117440 n^{4} + 365884416 n^{2} - 132710400\right)}\] 17 | \[\operatorname{Wg}([7, 1, 1]) = \frac{132 n^{2} - 2442}{n \left(n^{16} - 156 n^{14} + 8982 n^{12} - 245492 n^{10} + 3379233 n^{8} - 22878648 n^{6} + 69735184 n^{4} - 83176704 n^{2} + 33177600\right)}\] 18 | \[\operatorname{Wg}([2, 2, 2, 2, 1]) = \frac{n^{10} - 81 n^{8} + 2892 n^{6} - 22924 n^{4} - 534288 n^{2} + 1128960}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 19 | \[\operatorname{Wg}([4, 2, 2, 1]) = \frac{- 5 n^{8} + 219 n^{6} - 6926 n^{4} + 181656 n^{2} - 23744}{n^{2} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 20 | \[\operatorname{Wg}([3, 3, 2, 1]) = \frac{- 4 n^{4} - 32 n^{2} - 7308}{n^{2} \left(n^{16} - 169 n^{14} + 10542 n^{12} - 306098 n^{10} + 4388813 n^{8} - 30463797 n^{6} + 94092244 n^{4} - 112879936 n^{2} + 45158400\right)}\] 21 | \[\operatorname{Wg}([6, 2, 1]) = \frac{42 n^{4} + 204 n^{2} + 7962}{n \left(n^{18} - 173 n^{16} + 11218 n^{14} - 348266 n^{12} + 5613205 n^{10} - 48019049 n^{8} + 215947432 n^{6} - 489248912 n^{4} + 496678144 n^{2} - 180633600\right)}\] 22 | \[\operatorname{Wg}([5, 3, 1]) = \frac{28 n^{8} + 483 n^{6} - 66843 n^{4} + 362012 n^{2} + 188160}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 23 | \[\operatorname{Wg}([4, 4, 1]) = \frac{25 n^{8} + 935 n^{6} - 89200 n^{4} + 736720 n^{2} - 376320}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 24 | \[\operatorname{Wg}([8, 1]) = \frac{- 429 n^{2} + 16016}{n^{2} \left(n^{16} - 204 n^{14} + 16422 n^{12} - 669188 n^{10} + 14739153 n^{8} - 173721912 n^{6} + 1017067024 n^{4} - 2483133696 n^{2} + 1625702400\right)}\] 25 | \[\operatorname{Wg}([3, 2, 2, 2]) = \frac{- 2 n^{8} + 78 n^{6} - 5388 n^{4} - 115648 n^{2} - 665280}{n^{2} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 26 | \[\operatorname{Wg}([5, 2, 2]) = \frac{14 n^{8} - 66 n^{6} + 26916 n^{4} + 288976 n^{2} - 376320}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 27 | \[\operatorname{Wg}([4, 3, 2]) = \frac{10 n^{6} + 270 n^{4} + 33320 n^{2} - 94080}{n^{3} \left(n^{18} - 205 n^{16} + 16626 n^{14} - 685610 n^{12} + 15408341 n^{10} - 188461065 n^{8} + 1190788936 n^{6} - 3500200720 n^{4} + 4108836096 n^{2} - 1625702400\right)}\] 28 | \[\operatorname{Wg}([7, 2]) = \frac{- 132 n^{4} - 1100 n^{2} - 109648}{n^{2} \left(n^{18} - 205 n^{16} + 16626 n^{14} - 685610 n^{12} + 15408341 n^{10} - 188461065 n^{8} + 1190788936 n^{6} - 3500200720 n^{4} + 4108836096 n^{2} - 1625702400\right)}\] 29 | \[\operatorname{Wg}([3, 3, 3]) = \frac{8 n^{8} + 248 n^{6} + 48992 n^{4} - 825408 n^{2} - 1128960}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 30 | \[\operatorname{Wg}([6, 3]) = \frac{- 84 n^{6} - 5964 n^{4} + 41664 n^{2} + 538944}{n^{2} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 31 | \[\operatorname{Wg}([5, 4]) = \frac{- 70 n^{6} - 8050 n^{4} + 142520 n^{2} - 1041600}{n^{2} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\] 32 | \[\operatorname{Wg}([9]) = \frac{1430}{n \left(n^{16} - 204 n^{14} + 16422 n^{12} - 669188 n^{10} + 14739153 n^{8} - 173721912 n^{6} + 1017067024 n^{4} - 2483133696 n^{2} + 1625702400\right)}\] 33 | -------------------------------------------------------------------------------- /PYTHON/WFG_source.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | """ 3 | Created on Tue Jun 26 07:55:00 2018 4 | 5 | @author: M 6 | """ 7 | import os.path 8 | # import pickle 9 | import math 10 | import copy 11 | import fractions 12 | from sympy import Symbol, simplify, fraction 13 | from sympy.utilities.iterables import ordered_partitions 14 | from functools import reduce 15 | 16 | 17 | ########## Character Table ########## 18 | 19 | # "PartsStocker(List).Dic" gives a dictionary of 20 | # {# of removed boxes: [[List expression of valid border strip,signature],...]} 21 | # where 22 | # signature = (-1)^{height} 23 | # 24 | class PartsStocker: 25 | def __init__(self,List): 26 | self.List = List 27 | Range = range(len(self.List)) 28 | # enumerating BSTs to be removed. 29 | Collections = [[0]*i + x for i in Range for x in self.BS_Finder(self.List[i:])] 30 | Col_indexed = [ [sum(x),x] for x in Collections] 31 | # making a dictionary. 32 | self.Dic = {i+1:[] for i in range(sum(self.List))} 33 | for x in Col_indexed: 34 | # finding out what are left after each removal. 35 | # appending information on hights of removed BS. 36 | Leftover = [[self.List[i] - x[1][i] for i in Range],\ 37 | (-1)**(len(Range) - x[1].count(0) -1)] 38 | # deleting 0's if some (bottom) rows vanish. 39 | L = [[x for x in Leftover[0] if x !=0],Leftover[1]] 40 | self.Dic[x[0]].append(L) 41 | 42 | 43 | 44 | # Input: List specifies Young diagram, 45 | # Output: is a list=[b_1,...,b_n] specifying all valid border strips 46 | # each b_j is a list b_j=[e_1,\ldots,e_n] specifying a border strip 47 | # entry e_i specifies how many boxes belong to the i-th row of the border strip 48 | # 49 | def BS_Finder(self,List): 50 | Length = len(List) 51 | # calculating the difference of neighboring rows. 52 | Gaps = [List[i] - (List + [0])[i+1] for i in range(Length)] 53 | 54 | # getting sizes of invalid border strips. The last one is redundent. 55 | Bad = [sum(Gaps[0:i+1]) +i+1 for i in range(Length)] 56 | 57 | # enumerating all possible border strips, including invalid ones. 58 | r = [0]*Length 59 | BSs = [] 60 | for i in range(Length): 61 | for j in range(Gaps[i]+1): 62 | r[i] += 1 63 | new = copy.copy(r) 64 | BSs.append(new) 65 | 66 | # removing invalid ones 67 | Ex = Bad[:-1][::-1] 68 | for i in Ex: 69 | del(BSs[i-1]) 70 | # removing another redundant border strip generated for some technical reason 71 | del(BSs[-1]) 72 | return BSs 73 | 74 | 75 | 76 | """ 77 | class Character: 78 | def __init__(self,ListA,ListB): 79 | self.ListA = ListA 80 | self.ListB = ListB 81 | # quoting the class:PartsStocker. 82 | Deco = PartsStocker(self.ListA).Dic[self.ListB[0]] 83 | # preparing for the calculation of character. 84 | # we have set the character of [][] is 1 to deal with the case 85 | # where the leftover is empty. 86 | Summand = [x[1] * Memo(sum(x[0])).Data[str(x[0])+str(self.ListB[1:])]\ 87 | for x in Deco] 88 | self.Char = sum(Summand) 89 | """ 90 | 91 | 92 | #print(Character([1],[1]).Char) 93 | 94 | class CharacterTableCalculator: 95 | def __init__(self,k): 96 | self.k = k 97 | Partitions= [p[::-1] for p in ordered_partitions(self.k)] 98 | self.Dic = {} 99 | M = [Memo(i).Data for i in range(self.k)] 100 | for p in Partitions: 101 | Deco = PartsStocker(p).Dic 102 | 103 | for q in Partitions: 104 | self.Dic[str(p)+str(q)] =sum(\ 105 | [x[1] * M[sum(x[0])][str(x[0])+str(q[1:])] for x in Deco[q[0]]]) 106 | 107 | 108 | 109 | """ 110 | def Character(ListA,ListB): 111 | # quoting the class:PartsStocker. 112 | 113 | # preparing for the calculation of character. 114 | # we have set the character of [][] is 1 to deal with the case 115 | # where the leftover is empty. 116 | Summand = [x[1] * Memo(sum(x[0])).Data[str(x[0])+str(self.ListB[1:])]\ 117 | for x in Deco] 118 | self.Char = sum(Summand) 119 | 120 | """ 121 | 122 | #print(CharacterTableCalculator(1).Dic) 123 | # 124 | 125 | 126 | 127 | 128 | # making instances of characters:Memo(k). 129 | # if there is a file, reads it. 130 | # otherwise, generates and writes down. 131 | class Memo: 132 | def __init__(self,k): 133 | self.k = k 134 | self.Data = {} 135 | if self.k == 0: 136 | self.Data.update({'[][]':1}) 137 | 138 | elif os.path.isfile('SGC/table{}.txt'.format(self.k)): 139 | with open("SGC/table{}.txt".format(self.k), 'r') as file: 140 | self.Data.update(eval(file.read())) 141 | 142 | # elif os.path.isfile('SGC/table{}.pkl'.format(self.k)): 143 | # with open("SGC/table{}.pkl".format(self.k), 'rb') as file: 144 | # self.Data.update(pickle.load(file)) 145 | 146 | else: 147 | if not os.path.exists('SGC'): 148 | os.makedirs('SGC') 149 | else: 150 | pass 151 | self.Data.update(CharacterTableCalculator(self.k).Dic) 152 | 153 | with open("SGC/table{}.txt".format(self.k), "w") as fp: 154 | fp.write(str(self.Data)) 155 | 156 | # with open("SGC/table{}.pkl".format(self.k), "wb") as fp: #Pickling 157 | # pickle.dump(self.Data, fp) 158 | 159 | #print(Memo(14).Data) 160 | 161 | 162 | #print(CharacterTableOrganizer(3)) 163 | 164 | class SGCTOganizer: 165 | def __init__(self,k): 166 | self.k = k 167 | if not os.path.exists('SGC'): 168 | os.makedirs('SGC') 169 | else: 170 | pass 171 | for i in range(self.k): 172 | Memo(i) 173 | self.Dic = Memo(self.k).Data 174 | 175 | #print(SGCTOganizer(8).Dic) 176 | 177 | ########## Schur Polynomial ########## 178 | 179 | class SchurPolyGenerator: 180 | def __init__(self,k): 181 | self.k = k 182 | 183 | # Generating partitions of k in the non-increasing order. 184 | self.YoungDiagram = [p[::-1] for p in ordered_partitions(self.k)] 185 | # Making n into a symbol. 186 | n= Symbol('n') 187 | # Maling a list of reciprocals of Schur polynomials for all Young Diagrams. 188 | # [[permutation, Schur poly],[,],...] 189 | self.ReciprocalList =\ 190 | { str(p): self.SchurPolyReciprocal(p,n) for p in self.YoungDiagram} 191 | 192 | # Generating reciprocal of a Schur polynomial for a fixed Young Diagram... 193 | # [permutation, Schur poly] 194 | def SchurPolyReciprocal(self,p,n): 195 | l = len(p) 196 | Pairs = [[x,y] for x in range(l) for y in range(l) if x

Weingarten functions for p = 10

Below are the values of the Weingarten function for permutation size p = 10. The input is given as a partition of p. You can also download them as a text file or as a python pickle file. 2 | 3 | \[\operatorname{Wg}([1, 1, 1, 1, 1, 1, 1, 1, 1, 1]) = \frac{n^{16} - 254 n^{14} + 25165 n^{12} - 1239500 n^{10} + 32153848 n^{8} - 432276152 n^{6} + 2813672556 n^{4} - 7651234224 n^{2} + 5272646400}{n^{2} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 4 | \[\operatorname{Wg}([2, 1, 1, 1, 1, 1, 1, 1, 1]) = \frac{- n^{16} + 238 n^{14} - 21637 n^{12} + 950092 n^{10} - 21109312 n^{8} + 229937848 n^{6} - 1117994460 n^{4} + 1912326192 n^{2} - 640120320}{n^{3} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 5 | \[\operatorname{Wg}([3, 1, 1, 1, 1, 1, 1, 1]) = \frac{2 n^{12} - 423 n^{10} + 32369 n^{8} - 1089246 n^{6} + 15489074 n^{4} - 72558096 n^{2} + 64340640}{n^{2} \left(n^{22} - 290 n^{20} + 34375 n^{18} - 2165240 n^{16} + 79072015 n^{14} - 1720307690 n^{12} + 22202281945 n^{10} - 165778645340 n^{8} + 687441512560 n^{6} - 1484941803840 n^{4} + 1469447599104 n^{2} - 526727577600\right)}\] 6 | \[\operatorname{Wg}([2, 2, 1, 1, 1, 1, 1, 1]) = \frac{n^{14} - 216 n^{12} + 17335 n^{10} - 648702 n^{8} + 11758858 n^{6} - 98555292 n^{4} + 324025776 n^{2} - 282683520}{n^{2} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 7 | \[\operatorname{Wg}([4, 1, 1, 1, 1, 1, 1]) = \frac{- 5 n^{14} + 1014 n^{12} - 74231 n^{10} + 2418048 n^{8} - 35416562 n^{6} + 216946488 n^{4} - 463655232 n^{2} + 182891520}{n^{3} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 8 | \[\operatorname{Wg}([3, 2, 1, 1, 1, 1, 1]) = \frac{- 2 n^{14} + 379 n^{12} - 25488 n^{10} + 752393 n^{8} - 10091572 n^{6} + 60012618 n^{4} - 117055368 n^{2} + 45722880}{n^{3} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 9 | \[\operatorname{Wg}([5, 1, 1, 1, 1, 1]) = \frac{14 n^{12} - 2576 n^{10} + 165298 n^{8} - 4487812 n^{6} + 50921388 n^{4} - 215849592 n^{2} + 209653920}{n^{2} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 10 | \[\operatorname{Wg}([2, 2, 2, 1, 1, 1, 1]) = \frac{- n^{14} + 188 n^{12} - 12891 n^{10} + 410170 n^{8} - 6341894 n^{6} + 41526972 n^{4} - 61588944 n^{2} - 91445760}{n^{3} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 11 | \[\operatorname{Wg}([4, 2, 1, 1, 1, 1]) = \frac{5 n^{8} - 516 n^{6} + 10750 n^{4} - 39048 n^{2} + 45504}{n^{2} \left(n^{20} - 241 n^{18} + 22566 n^{16} - 1059506 n^{14} + 27156221 n^{12} - 389652861 n^{10} + 3109291756 n^{8} - 13423349296 n^{6} + 29697397056 n^{4} - 29769348096 n^{2} + 10749542400\right)}\] 12 | \[\operatorname{Wg}([3, 3, 1, 1, 1, 1]) = \frac{4 n^{10} - 576 n^{8} + 24152 n^{6} - 250848 n^{4} - 1166796 n^{2} + 4780944}{n^{2} \left(n^{22} - 295 n^{20} + 35805 n^{18} - 2331395 n^{16} + 89233595 n^{14} - 2075021445 n^{12} + 29384965375 n^{10} - 248059321345 n^{8} + 1187212035240 n^{6} - 2923067275920 n^{4} + 3141654729984 n^{2} - 1185137049600\right)}\] 13 | \[\operatorname{Wg}([6, 1, 1, 1, 1]) = \frac{- 42 n^{12} + 6828 n^{10} - 370614 n^{8} + 8018976 n^{6} - 67613724 n^{4} + 190353456 n^{2} - 91445760}{n^{3} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 14 | \[\operatorname{Wg}([3, 2, 2, 1, 1, 1]) = \frac{2 n^{10} - 287 n^{8} + 14167 n^{6} - 336634 n^{4} + 4176336 n^{2} - 8162784}{n^{2} \left(n^{22} - 290 n^{20} + 34375 n^{18} - 2165240 n^{16} + 79072015 n^{14} - 1720307690 n^{12} + 22202281945 n^{10} - 165778645340 n^{8} + 687441512560 n^{6} - 1484941803840 n^{4} + 1469447599104 n^{2} - 526727577600\right)}\] 15 | \[\operatorname{Wg}([5, 2, 1, 1, 1]) = \frac{- 14 n^{12} + 1832 n^{10} - 70906 n^{8} + 859012 n^{6} - 1616340 n^{4} - 26268624 n^{2} + 45722880}{n^{3} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 16 | \[\operatorname{Wg}([4, 3, 1, 1, 1]) = \frac{- 10 n^{12} + 1009 n^{10} - 11372 n^{8} - 927595 n^{6} + 18341952 n^{4} - 79456464 n^{2} + 45722880}{n^{3} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 17 | \[\operatorname{Wg}([7, 1, 1, 1]) = \frac{132 n^{8} - 16863 n^{6} + 619311 n^{4} - 6468132 n^{2} + 9690912}{n^{2} \left(n^{22} - 290 n^{20} + 34375 n^{18} - 2165240 n^{16} + 79072015 n^{14} - 1720307690 n^{12} + 22202281945 n^{10} - 165778645340 n^{8} + 687441512560 n^{6} - 1484941803840 n^{4} + 1469447599104 n^{2} - 526727577600\right)}\] 18 | \[\operatorname{Wg}([2, 2, 2, 2, 1, 1]) = \frac{n^{10} - 90 n^{8} + 3285 n^{6} - 31240 n^{4} - 182916 n^{2} + 3039120}{n^{2} \left(n^{22} - 235 n^{20} + 21945 n^{18} - 1070135 n^{16} + 30070535 n^{14} - 507441585 n^{12} + 5208789715 n^{10} - 32236641085 n^{8} + 116304291180 n^{6} - 228440781360 n^{4} + 213713826624 n^{2} - 74071065600\right)}\] 19 | \[\operatorname{Wg}([4, 2, 2, 1, 1]) = \frac{- 5 n^{6} + 203 n^{4} - 6494 n^{2} + 89576}{n \left(n^{20} - 226 n^{18} + 19911 n^{16} - 890936 n^{14} + 22052111 n^{12} - 308972586 n^{10} + 2428036441 n^{8} - 10384313116 n^{6} + 22845473136 n^{4} - 22831523136 n^{2} + 8230118400\right)}\] 20 | \[\operatorname{Wg}([3, 3, 2, 1, 1]) = \frac{- 4 n^{10} + 152 n^{8} - 7870 n^{6} + 217548 n^{4} - 1442826 n^{2} + 1428840}{n^{3} \left(n^{22} - 235 n^{20} + 21945 n^{18} - 1070135 n^{16} + 30070535 n^{14} - 507441585 n^{12} + 5208789715 n^{10} - 32236641085 n^{8} + 116304291180 n^{6} - 228440781360 n^{4} + 213713826624 n^{2} - 74071065600\right)}\] 21 | \[\operatorname{Wg}([6, 2, 1, 1]) = \frac{42 n^{6} - 1230 n^{4} + 8046 n^{2} - 75978}{n^{2} \left(n^{20} - 231 n^{18} + 21021 n^{16} - 986051 n^{14} + 26126331 n^{12} - 402936261 n^{10} + 3597044671 n^{8} - 17848462401 n^{6} + 44910441576 n^{4} - 48799015056 n^{2} + 18517766400\right)}\] 22 | \[\operatorname{Wg}([5, 3, 1, 1]) = \frac{28 n^{6} + 665 n^{4} - 43113 n^{2} + 104580}{n^{2} \left(n^{20} - 226 n^{18} + 19911 n^{16} - 890936 n^{14} + 22052111 n^{12} - 308972586 n^{10} + 2428036441 n^{8} - 10384313116 n^{6} + 22845473136 n^{4} - 22831523136 n^{2} + 8230118400\right)}\] 23 | \[\operatorname{Wg}([4, 4, 1, 1]) = \frac{25 n^{8} + 870 n^{6} - 69823 n^{4} + 772560 n^{2} - 1369872}{n^{2} \left(n^{22} - 235 n^{20} + 21945 n^{18} - 1070135 n^{16} + 30070535 n^{14} - 507441585 n^{12} + 5208789715 n^{10} - 32236641085 n^{8} + 116304291180 n^{6} - 228440781360 n^{4} + 213713826624 n^{2} - 74071065600\right)}\] 24 | \[\operatorname{Wg}([8, 1, 1]) = \frac{- 429 n^{2} + 11869}{n \left(n^{18} - 222 n^{16} + 19023 n^{14} - 814844 n^{12} + 18792735 n^{10} - 233801646 n^{8} + 1492829857 n^{6} - 4412993688 n^{4} + 5193498384 n^{2} - 2057529600\right)}\] 25 | \[\operatorname{Wg}([3, 2, 2, 2, 1]) = \frac{- 2 n^{12} + 219 n^{10} - 11766 n^{8} + 221027 n^{6} + 2702418 n^{4} - 26741016 n^{2} + 45722880}{n^{3} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 26 | \[\operatorname{Wg}([5, 2, 2, 1]) = \frac{14 n^{10} - 1004 n^{8} + 39718 n^{6} - 1098496 n^{4} - 1873512 n^{2} + 62687520}{n^{2} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 27 | \[\operatorname{Wg}([4, 3, 2, 1]) = \frac{10 n^{6} + 137 n^{4} + 28932 n^{2} - 133344}{n^{2} \left(n^{20} - 241 n^{18} + 22566 n^{16} - 1059506 n^{14} + 27156221 n^{12} - 389652861 n^{10} + 3109291756 n^{8} - 13423349296 n^{6} + 29697397056 n^{4} - 29769348096 n^{2} + 10749542400\right)}\] 28 | \[\operatorname{Wg}([7, 2, 1]) = \frac{- 132 n^{4} + 121 n^{2} - 79684}{n \left(n^{20} - 241 n^{18} + 22566 n^{16} - 1059506 n^{14} + 27156221 n^{12} - 389652861 n^{10} + 3109291756 n^{8} - 13423349296 n^{6} + 29697397056 n^{4} - 29769348096 n^{2} + 10749542400\right)}\] 29 | \[\operatorname{Wg}([3, 3, 3, 1]) = \frac{8 n^{10} - 292 n^{8} + 39704 n^{6} - 3139956 n^{4} + 45793368 n^{2} - 125429472}{n^{2} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 30 | \[\operatorname{Wg}([6, 3, 1]) = \frac{- 84 n^{10} - 618 n^{8} + 318360 n^{6} - 3678402 n^{4} - 3413016 n^{2} + 45722880}{n^{3} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 31 | \[\operatorname{Wg}([5, 4, 1]) = \frac{- 70 n^{10} - 3584 n^{8} + 539266 n^{6} - 10376716 n^{4} + 64515024 n^{2} - 45722880}{n^{3} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 32 | \[\operatorname{Wg}([9, 1]) = \frac{1430 n^{2} - 72072}{n^{2} \left(n^{18} - 285 n^{16} + 32946 n^{14} - 1999370 n^{12} + 68943381 n^{10} - 1367593305 n^{8} + 15088541896 n^{6} - 84865562640 n^{4} + 202759531776 n^{2} - 131681894400\right)}\] 33 | \[\operatorname{Wg}([2, 2, 2, 2, 2]) = \frac{- n^{12} + 114 n^{10} - 6645 n^{8} + 52480 n^{6} - 3861804 n^{4} - 82670544 n^{2} + 457228800}{n^{3} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 34 | \[\operatorname{Wg}([4, 2, 2, 2]) = \frac{5 n^{10} - 300 n^{8} + 23625 n^{6} + 255670 n^{4} + 8490600 n^{2} - 32840640}{n^{2} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 35 | \[\operatorname{Wg}([3, 3, 2, 2]) = \frac{4 n^{10} - 200 n^{8} + 22372 n^{6} + 390960 n^{4} - 3860496 n^{2} - 72031680}{n^{2} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 36 | \[\operatorname{Wg}([6, 2, 2]) = \frac{- 42 n^{10} + 1092 n^{8} - 125034 n^{6} - 1672272 n^{4} + 28165536 n^{2} - 91445760}{n^{3} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 37 | \[\operatorname{Wg}([5, 3, 2]) = \frac{- 28 n^{10} - 529 n^{8} - 114478 n^{6} + 868119 n^{4} + 10254276 n^{2} + 45722880}{n^{3} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 38 | \[\operatorname{Wg}([4, 4, 2]) = \frac{- 25 n^{10} - 950 n^{8} - 109025 n^{6} + 2119520 n^{4} - 32854320 n^{2} + 91445760}{n^{3} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 39 | \[\operatorname{Wg}([8, 2]) = \frac{429 n^{4} - 429 n^{2} + 720720}{n^{2} \left(n^{20} - 286 n^{18} + 33231 n^{16} - 2032316 n^{14} + 70942751 n^{12} - 1436536686 n^{10} + 16456135201 n^{8} - 99954104536 n^{6} + 287625094416 n^{4} - 334441426176 n^{2} + 131681894400\right)}\] 40 | \[\operatorname{Wg}([4, 3, 3]) = \frac{- 20 n^{10} - 800 n^{8} - 212660 n^{6} + 7008840 n^{4} - 41631840 n^{2} - 91445760}{n^{3} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 41 | \[\operatorname{Wg}([7, 3]) = \frac{264 n^{6} + 20856 n^{4} + 1056 n^{2} - 7672896}{n^{2} \left(n^{22} - 290 n^{20} + 34375 n^{18} - 2165240 n^{16} + 79072015 n^{14} - 1720307690 n^{12} + 22202281945 n^{10} - 165778645340 n^{8} + 687441512560 n^{6} - 1484941803840 n^{4} + 1469447599104 n^{2} - 526727577600\right)}\] 42 | \[\operatorname{Wg}([6, 4]) = \frac{210 n^{8} + 29316 n^{6} - 920094 n^{4} + 10458504 n^{2} - 22389696}{n^{2} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 43 | \[\operatorname{Wg}([5, 5]) = \frac{196 n^{8} + 32536 n^{6} - 1189916 n^{4} + 20208384 n^{2} - 150413760}{n^{2} \left(n^{24} - 299 n^{22} + 36985 n^{20} - 2474615 n^{18} + 98559175 n^{16} - 2431955825 n^{14} + 37685051155 n^{12} - 365599182845 n^{10} + 2179449320620 n^{8} - 7671915416880 n^{6} + 14833923833664 n^{4} - 13751755969536 n^{2} + 4740548198400\right)}\] 44 | \[\operatorname{Wg}([10]) = - \frac{4862}{n \left(n^{18} - 285 n^{16} + 32946 n^{14} - 1999370 n^{12} + 68943381 n^{10} - 1367593305 n^{8} + 15088541896 n^{6} - 84865562640 n^{4} + 202759531776 n^{2} - 131681894400\right)}\] 45 | -------------------------------------------------------------------------------- /PYTHON/IHU_source.py: -------------------------------------------------------------------------------- 1 | # -*- coding: utf-8 -*- 2 | """ 3 | a copy of "average_graph_tensor12" with a corection in Calc.Reconnection. 4 | 5 | @author: M 6 | """ 7 | import os.path 8 | from sympy import symbols, simplify 9 | from sympy.combinatorics import Permutation 10 | from itertools import groupby 11 | import copy 12 | import networkx as nx 13 | import itertools 14 | import matplotlib.pyplot as plt 15 | import pickle 16 | 17 | 18 | 19 | from WFG_source import * 20 | 21 | 22 | # preparing several things for later use. 23 | class Prep: 24 | def __init__(self,Edges,Weight,RM_List): 25 | self.Edges = Edges 26 | self.Weight = Weight 27 | self.RM_List = RM_List 28 | 29 | self.RM_Number = len(RM_List) #!!! 30 | 31 | 32 | # extracting lims of matrices. 33 | M_Lims = [ x[y] for x in Edges for y in [0,1]] 34 | # selecting lims of random matrices. 35 | RM_Lims = [x for x in M_Lims for rm in RM_List if x[0] == rm[0] or x[0] == rm[0]+'*'] 36 | 37 | # identifying matrices from the above list of lims. 38 | Matrices = [x[0:2] for x in M_Lims] 39 | 40 | 41 | # for each random matrix set explicitly, we collect and sort random matrices identified from the lims. 42 | RM1 = [sorted([x for x in Matrices if x[0] == rm[0]],key = lambda t: t[1]) 43 | for rm in RM_List] 44 | RM2 = [sorted([x for x in Matrices if x[0] == rm[0] +'*'],key = lambda t: t[1]) 45 | for rm in RM_List] 46 | 47 | # putting the same matrices together. 48 | self.RM1_grouped = [[x for x,y in groupby(rm1)] for rm1 in RM1] #!!!(?) 49 | RM2_grouped = sorted([[x for x,y in groupby(rm2)] for rm2 in RM2]) 50 | 51 | # listing random matrices in the calculus. 52 | self.RMs = [y[0] for x in self.RM1_grouped for y in x]+[y[0] for x in RM2_grouped for y in x] #!!! 53 | 54 | # relabeling the random matrices for the sake of Weingarten calculus. 55 | RM1_Labels = [[ [rm1g[i][0:2] ,[rm1g[i][0],i]] 56 | for i in range(len(rm1g))] for rm1g in self.RM1_grouped] 57 | RM2_Labels = [[ [rm2g[i][0:2] ,[rm2g[i][0],i]] 58 | for i in range(len(rm2g))] for rm2g in RM2_grouped] 59 | 60 | # making a dictionary for relabeling. 61 | Translation = [ [x[0][0]+str(x[0][1]) ,x[1][0:2]] for i in range(self.RM_Number) for x in RM1_Labels[i] ]+\ 62 | [ [x[0][0]+str(x[0][1]) ,x[1][0:2]] for i in range(self.RM_Number) for x in RM2_Labels[i] ] 63 | Translation_inv = [ [x[1][0]+str(x[1][1]) ,x[0][0]+str(x[0][1])] for i in range(self.RM_Number) for x in RM1_Labels[i] ]+\ 64 | [ [x[1][0]+str(x[1][1]) ,x[0][0]+str(x[0][1])] for i in range(self.RM_Number) for x in RM2_Labels[i] ] 65 | self.Label_Dic = dict(Translation) #!!! 66 | self.Label_Dic_inv = dict(Translation_inv) #!!! 67 | 68 | # listing all possible limes; 69 | # for each random matrix set explicitly, 70 | # generating all the lims of random matrices in the calculus. 71 | RM_Lims_generated1 =\ 72 | [[x+['L',y] for x in self.RM1_grouped[i] for y in range(len(RM_List[i][1]))]\ 73 | +[x+['R',y] for x in self.RM1_grouped[i] for y in range(len(RM_List[i][2]))] for i in range(self.RM_Number)] 74 | RM_Lims_generated2 =\ 75 | [[x+['L',y] for x in RM2_grouped[i] for y in range(len(RM_List[i][2]))]\ 76 | +[x+['R',y] for x in RM2_grouped[i] for y in range(len(RM_List[i][1]))] for i in range(self.RM_Number)] 77 | RM_Lims_generated = [[x for x in RM_Lims_generated1[i] ] +\ 78 | [x for x in RM_Lims_generated2[i] ] for i in range(self.RM_Number)] 79 | RM_Lims_generated_flat = [x for i in range(self.RM_Number) for x in RM_Lims_generated[i]] 80 | 81 | self.RMLg_String = [ [x[0]+str(x[1])+x[2]+str(x[3]) for x in RM_Lims_generated[i]] for i in range(self.RM_Number)] #!!! 82 | 83 | # listing lims of random matrices without *. 84 | self.RMLg1_relabeled = [[self.Label_Dic[x[0]+str(x[1])] +x[2:4] for x in RM_Lims_generated1[i]] for i in range(self.RM_Number)] #!!! 85 | 86 | # finding empty lims. 87 | RM_Lims_empty = [x for x in RM_Lims_generated_flat if x not in RM_Lims] 88 | 89 | # adding extra edges for each empty lim. 90 | Edges_extra = [[x,['@'+x[0]] + x[1:4]] for x in RM_Lims_empty] 91 | 92 | # preparing edges for Weingarten calculus. 93 | Edges_Graph = Edges + Edges_extra 94 | 95 | # treating cases which gives 0. 96 | for i in range(self.RM_Number): 97 | if [ len(x) for x in self.RM1_grouped[i]] != [ len(x) for x in RM2_grouped[i]]: 98 | Edges_Graph.clear() 99 | Weight = 0 100 | else: 101 | pass 102 | 103 | # preparing a graph for Weingarten calculus; 104 | self.G = nx.MultiGraph() 105 | for x in Edges_Graph: 106 | a = x[0][0]+str(x[0][1])+x[0][2]+str(x[0][3]) 107 | b = x[1][0]+str(x[1][1])+x[1][2]+str(x[1][3]) 108 | self.G.add_edge(a,b,info = [a,b]) 109 | 110 | # storing the original lists, aftet changing them into strings. 111 | self.G.nodes[a]['original'] = x[0] 112 | self.G.nodes[b]['original'] = x[1] 113 | 114 | 115 | 116 | 117 | # loading Weingarten functions to be used; n is the default symbolic variable for the dimension. 118 | class WGF: 119 | def __init__(self,SDs): 120 | self.SDs = SDs 121 | Sizes = [x[0] for x in self.SDs] 122 | Dims = [x[1] for x in self.SDs] 123 | n = symbols('n') 124 | self.Dic ={} 125 | self.Addressbook ={} 126 | for i in range(len(SDs)): 127 | 128 | if not os.path.isfile('Weingarten/functions{}.txt'.format(Sizes[i])): 129 | Photocopy = WFs(Sizes[i]).wfs 130 | else: 131 | with open('Weingarten/functions{}.txt'.format(Sizes[i]), 'r') as file: 132 | n= Symbol('n') 133 | Photocopy = eval(file.read()) 134 | 135 | # if not os.path.isfile('Weingarten/functions{}.pkl'.format(Sizes[i])): 136 | # Photocopy = WFs(Sizes[i]).wfs 137 | # else: 138 | # with open('Weingarten/functions{}.pkl'.format(Sizes[i]), 'rb') as file: 139 | # Photocopy = pickle.load(file)# Unpickling 140 | 141 | self.Dic['wg{}'.format(Sizes[i])] = [[p[0],p[1].subs(n,Dims[i])] for p in Photocopy] 142 | # Here, substituting n by t the actual size of matrix. 143 | self.Addressbook['wg{}'.format(Sizes[i])] = [p[0] for p in Photocopy] 144 | 145 | 146 | 147 | 148 | # averaging each diagram. 149 | #Calc.output = [[G,W],...] 150 | class Calc: 151 | def __init__(self,Edges,Weight,RM_List): 152 | self.Edges = Edges 153 | self.Weight = Weight 154 | self.RM_List = RM_List 155 | self.PRE = Prep(Edges,Weight,RM_List) 156 | SDs = [[len(self.PRE.RM1_grouped[i]),self.RM_List[i][3]] for i in range(self.PRE.RM_Number)] 157 | self.WGF_all = WGF(SDs) 158 | self.Output = self.Calculator() 159 | 160 | # graphical calculus for one RM. 161 | def Reconnection(self,Graph,a,b,RM_ID): 162 | w = 1 163 | G = copy.deepcopy(Graph) 164 | 165 | Average_Connections =\ 166 | [ [x, [x[0]+'*',Permutation(a)(x[1]),'R',x[3]]]\ 167 | for x in self.PRE.RMLg1_relabeled[RM_ID] if x[2] == 'L'] +\ 168 | [ [x, [x[0]+'*',Permutation(b)(x[1]),'L',x[3]]]\ 169 | for x in self.PRE.RMLg1_relabeled[RM_ID] if x[2] == 'R'] 170 | 171 | # making node names in strings and adding new edes to the graph. 172 | Average_Connections_List =\ 173 | [[self.PRE.Label_Dic_inv[x[0][0]+str(x[0][1])]+x[0][2]+str(x[0][3]),\ 174 | self.PRE.Label_Dic_inv[x[1][0]+str(x[1][1])]+x[1][2]+str(x[1][3])] for x in Average_Connections] 175 | G.add_edges_from(Average_Connections_List) 176 | 177 | # tidying up: 178 | for y in self.PRE.RMLg_String[RM_ID]: # for each RM lim 179 | neighbour = list(G.neighbors(y)) 180 | if len(neighbour) == 2: #case where the lim is connected to two nodes. 181 | G.remove_node(y) 182 | G.add_edge(neighbour[0],neighbour[1], info = neighbour) 183 | 184 | elif G.degree(y) == 2: 185 | G.remove_node(y) 186 | 187 | else: # case the node is a pendant. 188 | if int(nx.__version__[0])<2: 189 | if G.node[y]['original'][0][-1] != '*' and G.node[y]['original'][2] == 'L' : 190 | side = 1 191 | elif G.node[y]['original'][0][-1] == '*' and G.node[y]['original'][2] == 'R': 192 | side = 1 193 | else: 194 | side = 2 195 | w *= self.RM_List[RM_ID][side][G.nodes[y]['original'][3]] 196 | G.remove_node(y) 197 | else: 198 | if G.nodes[y]['original'][0][-1] != '*' and G.nodes[y]['original'][2] == 'L' : 199 | side = 1 200 | elif G.nodes[y]['original'][0][-1] == '*' and G.nodes[y]['original'][2] == 'R': 201 | side = 1 202 | else: 203 | side = 2 204 | w *= self.RM_List[RM_ID][side][G.nodes[y]['original'][3]] 205 | G.remove_node(y) 206 | 207 | 208 | #Multiplying Weingarten function. 209 | t = len(self.PRE.RM1_grouped[RM_ID]) 210 | b_inv = [b.index(i) for i in range(t)] 211 | c = (Permutation(a) * Permutation(b_inv)).full_cyclic_form 212 | Cycle = sorted([ len(cycle) for cycle in c],reverse=True) 213 | Address = self.WGF_all.Addressbook['wg{}'.format(t)].index(Cycle) 214 | w *= self.WGF_all.Dic['wg{}'.format(t)][Address][1] 215 | 216 | return G, simplify(w) 217 | 218 | 219 | # calculating the average of a diagram over a RM. 220 | def Average(self,Graph,Weight,RM_ID): 221 | Average = [] 222 | Perms = list(itertools.permutations(range(len(self.PRE.RM1_grouped[RM_ID])))) 223 | for pair in itertools.product(Perms,Perms): 224 | Ave_each = self.Reconnection(Graph,pair[0],pair[1],RM_ID) 225 | Weight_out = simplify(Weight * Ave_each[1]) 226 | Average.append([Ave_each[0],Weight_out]) 227 | return Average 228 | 229 | 230 | # calculating the average over RMs selected. 231 | def Calculator(self): 232 | if self.PRE.RM_Number == 0: # case where no average calculation is made. 233 | return [[self.PRE.G, self.PRE.Weight]] 234 | else: 235 | pass 236 | G = copy.deepcopy(self.PRE.G) 237 | W = copy.deepcopy(self.PRE.Weight) 238 | GW_List = copy.deepcopy([[G,W]]) # the initial list. 239 | # averaging a list of graphs over each RM one by one. 240 | for num in range(self.PRE.RM_Number): 241 | l = len(GW_List) # after first RM, the list may be longer than one. 242 | for i in range(l): 243 | GW_List += self.Average(GW_List[i][0],GW_List[i][1],num) 244 | del GW_List[:l] 245 | 246 | return GW_List 247 | 248 | 249 | 250 | # drawing edges. 251 | def visualizeGraphList0(EW): 252 | Edges = EW[0] 253 | Weight = EW[1] 254 | # sorting edges first within each edge and then according to the first element. 255 | Edges_sorted = sorted([ sorted(x, key = lambda s: s[0]) for x in Edges],\ 256 | key = lambda t: t[0]) 257 | #print(Edges_sorted) 258 | # making a list of new edges with nodes being matrices; 259 | #[new edge, label, concatinated nodes of new edge, original edge]; 260 | Edges_labeled = [[[x[0][0]+str(x[0][1]), x[1][0]+str(x[1][1])] ,\ 261 | '[' + x[0][2]+str(x[0][3])+':'+ x[1][2]+str(x[1][3])+']' ,\ 262 | x[0][0]+str(x[0][1]) +x[1][0]+str(x[1][1]) , x] for x in Edges_sorted] 263 | #print(Edges_labeled) 264 | 265 | 266 | 267 | # sorting wrt concatinated nodes of new edges. 268 | ### not sure if we really need this because it may not change anything. 269 | Edges_new = sorted(Edges_labeled, key = lambda t: t[2]) 270 | #print(Edges_new) 271 | # if a node has @, we get the original name back, 272 | #because we do not put @s' together... Endpoints are separated. 273 | for x in Edges_new: 274 | for i in [0,1]: 275 | if x[0][i][0] == '@': 276 | x[0][i] += x[3][i][2] + str(x[3][i][3]) 277 | else: 278 | pass 279 | #print(Edges_new) 280 | # adding technical nodes for drawing in case of self loops. 281 | for x in Edges_new: 282 | if x[0][0] == x[0][1]: 283 | x[0][0] += ' Loop' 284 | else: 285 | pass 286 | 287 | # putting double edges together by grouping only w.r.t. new edges of string-format. 288 | # [[new edge, [[connection],[connection],,,]...],] 289 | Compressed = [ [key, [a[1] for a in group]] for key, group in groupby(Edges_new, lambda x: x[0])] 290 | # making a new list of information for each edge because double-edges were compressed. 291 | # [[new edge, '[connection][connection]...'],] 292 | Compressed2 = [] 293 | for x in Compressed: 294 | y = '' 295 | # finding out all connections between the same two matrices. 296 | for z in x[1]: 297 | y += z 298 | Compressed2.append([x[0],y]) 299 | 300 | # generating a directed graph from the above new edges and setting the color to be red. 301 | H = nx.DiGraph() 302 | for x in Compressed2: 303 | H.add_edge(x[0][0],x[0][1],name = x[1],color='r') 304 | # putting that color information into the list. 305 | Colors_Edges = [H[u][v]['color'] for u,v in H.edges()] 306 | 307 | # defining colors of nodes. 308 | Color_nodes = [] 309 | for g in H: 310 | if g[-4:] == 'Loop': 311 | Color_nodes.append('orange') 312 | else: 313 | Color_nodes.append('yellow') 314 | 315 | # drawing. 316 | pos = nx.circular_layout(H) 317 | nx.draw(H, pos, with_labels = True, node_color = Color_nodes, edge_color=Colors_Edges) 318 | nx.draw_networkx_labels(H, pos) 319 | edge_labels = nx.get_edge_attributes(H,'name') 320 | nx.draw_networkx_edge_labels(H, pos, edge_labels) 321 | plt.show() 322 | print(Weight) 323 | 324 | # making the drawing function polymorphic: accepting list of diagrams automatically. 325 | def visualizeTN(EW): 326 | if isinstance(EW[-1], list): 327 | for ew in EW: 328 | visualizeGraphList0(ew) 329 | else: 330 | visualizeGraphList0(EW) 331 | 332 | 333 | 334 | def integrateHaarUnitary(EWs, RM_List): 335 | 336 | ##### making this function polymorphic ##### 337 | if not isinstance(EWs[-1], list): 338 | EWs = [EWs] 339 | else: 340 | pass 341 | if RM_List == []: 342 | pass 343 | elif not isinstance(RM_List[-1], list): 344 | RM_List = [RM_List] 345 | else: 346 | pass 347 | ##### up to here ##### 348 | 349 | ##### mathematica to python ##### 350 | 351 | # making a dictionary for translations between Python and Mathematica codes. 352 | Dic_PyMa = {} 353 | Dic_PyMa["L"] = "out" 354 | Dic_PyMa["R"] = "in" 355 | Dic_PyMa["in"] = "R" 356 | Dic_PyMa["out"] = "L" 357 | 358 | # changing out-in format to L-R format, and changing numbering system to mathematica to python 359 | EWs_p = [[[[z[0:2]+[Dic_PyMa[z[2]],z[3]-1] for z in y] for y in x[0]] ,x[1]]\ 360 | for x in EWs] 361 | 362 | # switching the order of L and R, in order to work like Mathematica package. 363 | RM_List = [[rm[0],rm[2],rm[1],rm[3]] for rm in RM_List] 364 | 365 | ##### up to here ##### 366 | 367 | # making a collection of graphs and weights. 368 | Collections = [] 369 | for x in EWs_p: 370 | Collections += Calc(x[0],x[1],RM_List).Output 371 | Collections_List = [ [list(c[0].edges()), c[1]] for c in Collections] 372 | #print("Collections_List",Collections_List) 373 | 374 | 375 | # making a dictionary to get the original form of input diagrams; 376 | # only for matrices left after the average, considering all graphs in Collections 377 | if int(nx.__version__[0])<2: 378 | Node_Dic = {y: x[0].node[y]['original'] for x in Collections for y in list(x[0])} 379 | else: 380 | Node_Dic = {y: x[0].nodes[y]['original'] for x in Collections for y in list(x[0])} 381 | #print("Node_Dic", Node_Dic) 382 | 383 | # making the graph data into list data. (EW = Edges and weight) 384 | Ave_EW = [[list(x[0].edges()),x[1]] for x in Collections] 385 | # sorting first within each edge, and second edges themselves w.r.t. the first element. 386 | Ave_EW_sorted =\ 387 | [[sorted([sorted(y) for y in x[0]], key = lambda t: t[0]),x[1]] for x in Ave_EW] 388 | 389 | # putting the same graphs into one group, and summing put their weights. 390 | Ave_EW_grouped = [] 391 | for x in Ave_EW_sorted: 392 | # finding the same graph(s) in the developing new list. 393 | # The number of such graphs should be one, though... 394 | Ind_same = [i for i in range(len(Ave_EW_grouped)) if Ave_EW_grouped[i][0] == x[0]] 395 | if len(Ind_same) >0: 396 | Ave_EW_grouped[Ind_same[0]][1] += x[1] 397 | else: 398 | Ave_EW_grouped.append(x) 399 | 400 | Ave_EW_grouped_tidy = [[x[0],simplify(x[1])] for x in Ave_EW_grouped] 401 | AEgt_List = [[[ [Node_Dic[x[0]],Node_Dic[x[1]]] for x in X[0] ] ,X[1]] for X in Ave_EW_grouped_tidy] 402 | 403 | ##### python to mathematica ##### 404 | 405 | # changing L-R format to out-in format, and changing numbering system from python to mathematica 406 | Ave_List_m = [[[[z[0:2]+[Dic_PyMa[z[2]],z[3]+1] for z in y] for y in x[0]] ,x[1]]\ 407 | for x in AEgt_List] 408 | 409 | ##### up to here ##### 410 | 411 | return Ave_List_m 412 | 413 | 414 | 415 | 416 | """ 417 | 418 | """ 419 | -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions11.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], (n**18 - 344*n**16 + 47305*n**14 - 3338030*n**12 + 129218848*n**10 - 2728819832*n**8 + 29688278076*n**6 - 149061471624*n**4 + 273312649440*n**2 - 38799129600)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[2, 1, 1, 1, 1, 1, 1, 1, 1, 1], (-n**16 + 326*n**14 - 41797*n**12 + 2688500*n**10 - 91884112*n**8 + 1635300968*n**6 - 13995701820*n**4 + 49708341936*n**2 - 48846551040)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[3, 1, 1, 1, 1, 1, 1, 1, 1], 2*(n**16 - 306*n**14 + 36085*n**12 - 2074152*n**10 + 60751048*n**8 - 871809792*n**6 + 5519618316*n**4 - 12470338800*n**2 + 2155507200)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[2, 2, 1, 1, 1, 1, 1, 1, 1], (n**16 - 302*n**14 + 35119*n**12 - 1992224*n**10 + 57820486*n**8 - 828604784*n**6 + 5240619624*n**4 - 11643104160*n**2 + 6923750400)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[4, 1, 1, 1, 1, 1, 1, 1], (-5*n**14 + 1428*n**12 - 153587*n**10 + 7783266*n**8 - 190872794*n**6 + 2119020876*n**4 - 9309500784*n**2 + 11672760960)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[3, 2, 1, 1, 1, 1, 1, 1], (-2*n**12 + 528*n**10 - 50834*n**8 + 2184936*n**6 - 41042804*n**4 + 275906256*n**2 - 294877440)/(n**2*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[5, 1, 1, 1, 1, 1, 1], 14*(n**14 - 264*n**12 + 25567*n**10 - 1121958*n**8 + 22416562*n**6 - 183771348*n**4 + 508690800*n**2 - 158630400)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[2, 2, 2, 1, 1, 1, 1, 1], (-n**14 + 272*n**12 - 27939*n**10 + 1373134*n**8 - 33834566*n**6 + 392961564*n**4 - 1655003664*n**2 + 1788168960)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[4, 2, 1, 1, 1, 1, 1], 5*(n**12 - 233*n**10 + 18774*n**8 - 614998*n**6 + 7273496*n**4 - 19792128*n**2 + 34836480)/(n**3*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[3, 3, 1, 1, 1, 1, 1], 4*(n**14 - 229*n**12 + 18002*n**10 - 569693*n**8 + 6534687*n**6 - 22616928*n**4 + 38406960*n**2 + 97977600)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[6, 1, 1, 1, 1, 1], (-42*n**12 + 10068*n**10 - 857334*n**8 + 31554696*n**6 - 492637644*n**4 + 2842232976*n**2 - 4405415040)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[3, 2, 2, 1, 1, 1, 1], 2*(n**14 - 234*n**12 + 20167*n**10 - 830664*n**8 + 17947366*n**6 - 188054412*n**4 + 559562256*n**2 - 65318400)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[5, 2, 1, 1, 1, 1], (-14*n**8 + 1902*n**6 - 57796*n**4 + 304788*n**2 + 1139760)/(n**2*(n**22 - 326*n**20 + 42511*n**18 - 2882036*n**16 + 111145711*n**14 - 2514183686*n**12 + 33325295041*n**10 - 253187957216*n**8 + 1061276784736*n**6 - 2307378836736*n**4 + 2291382432000*n**2 - 823011840000))], [[4, 3, 1, 1, 1, 1], (-10*n**10 + 1702*n**8 - 71504*n**6 - 521212*n**4 + 50868624*n**2 - 167034240)/(n**2*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[7, 1, 1, 1, 1], 132*(n**12 - 211*n**10 + 15119*n**8 - 438257*n**6 + 4918740*n**4 - 18365472*n**2 + 10886400)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[2, 2, 2, 2, 1, 1, 1], (n**14 - 236*n**12 + 21033*n**10 - 889090*n**8 + 16846604*n**6 - 75657384*n**4 - 608866848*n**2 + 2482099200)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[4, 2, 2, 1, 1, 1], (-5*n**10 + 917*n**8 - 58276*n**6 + 1799668*n**4 - 31856304*n**2 + 168658560)/(n**2*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[3, 3, 2, 1, 1, 1], (-4*n**10 + 708*n**8 - 45506*n**6 + 1745352*n**4 - 47293830*n**2 + 468711360)/(n**2*(n**24 - 395*n**22 + 65305*n**20 - 5911895*n**18 + 322373095*n**16 - 10998380945*n**14 + 236887109875*n**12 - 3186555858845*n**10 + 25993144169740*n**8 - 121644270799920*n**6 + 295448382321984*n**4 - 315350610048000*n**2 + 118513704960000))], [[6, 2, 1, 1, 1], 6*(7*n**12 - 1207*n**10 + 64643*n**8 - 1240589*n**6 + 8910450*n**4 - 7330104*n**2 + 21772800)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[5, 3, 1, 1, 1], 7*(4*n**12 - 499*n**10 + 3908*n**8 + 1064929*n**6 - 27354462*n**4 + 155108520*n**2 - 177292800)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[4, 4, 1, 1, 1], (25*n**12 - 2620*n**10 - 66379*n**8 + 11938846*n**6 - 279926736*n**4 + 1516001184*n**2 - 653184000)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[8, 1, 1, 1], (-429*n**8 + 71643*n**6 - 3540966*n**4 + 51552072*n**2 - 84839040)/(n**2*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[3, 2, 2, 2, 1, 1], (-2*n**8 + 198*n**6 - 9012*n**4 + 58928*n**2 + 2211840)/(n**2*(n**22 - 309*n**20 + 38346*n**18 - 2496714*n**16 + 93362181*n**14 - 2065172529*n**12 + 26954076096*n**10 - 202726676064*n**8 + 844445285376*n**6 - 1829024944384*n**4 + 1812607488000*n**2 - 650280960000))], [[5, 2, 2, 1, 1], 2*(7*n**10 - 467*n**8 + 18200*n**6 - 327236*n**4 - 62960*n**2 + 5644800)/(n**3*(n**24 - 318*n**22 + 41127*n**20 - 2841828*n**18 + 115832607*n**16 - 2905432158*n**14 + 45540628857*n**12 - 445313360928*n**10 + 2668985369952*n**8 - 9429032512768*n**6 + 18273831987456*n**4 - 16963748352000*n**2 + 5852528640000))], [[4, 3, 2, 1, 1], 2*(5*n**8 - 153*n**6 + 12084*n**4 - 483008*n**2 + 537600)/(n**3*(n**22 - 309*n**20 + 38346*n**18 - 2496714*n**16 + 93362181*n**14 - 2065172529*n**12 + 26954076096*n**10 - 202726676064*n**8 + 844445285376*n**6 - 1829024944384*n**4 + 1812607488000*n**2 - 650280960000))], [[7, 2, 1, 1], (-132*n**6 + 6094*n**4 - 105754*n**2 + 1995840)/(n**2*(n**22 - 314*n**20 + 39871*n**18 - 2682344*n**16 + 105103231*n**14 - 2485019234*n**12 + 35600551921*n**10 - 302911153244*n**8 + 1457340756976*n**6 - 3599669484864*n**4 + 3875154048000*n**2 - 1463132160000))], [[3, 3, 3, 1, 1], 4*(2*n**8 - 46*n**6 + 10007*n**4 - 448380*n**2 + 806400)/(n**3*(n**22 - 309*n**20 + 38346*n**18 - 2496714*n**16 + 93362181*n**14 - 2065172529*n**12 + 26954076096*n**10 - 202726676064*n**8 + 844445285376*n**6 - 1829024944384*n**4 + 1812607488000*n**2 - 650280960000))], [[6, 3, 1, 1], (-84*n**6 - 2076*n**4 + 208848*n**2 - 140160)/(n**2*(n**22 - 309*n**20 + 38346*n**18 - 2496714*n**16 + 93362181*n**14 - 2065172529*n**12 + 26954076096*n**10 - 202726676064*n**8 + 844445285376*n**6 - 1829024944384*n**4 + 1812607488000*n**2 - 650280960000))], [[5, 4, 1, 1], (-70*n**8 - 4158*n**6 + 408324*n**4 - 5866448*n**2 + 16007040)/(n**2*(n**24 - 318*n**22 + 41127*n**20 - 2841828*n**18 + 115832607*n**16 - 2905432158*n**14 + 45540628857*n**12 - 445313360928*n**10 + 2668985369952*n**8 - 9429032512768*n**6 + 18273831987456*n**4 - 16963748352000*n**2 + 5852528640000))], [[9, 1, 1], 286*(5*n**2 - 194)/(n*(n**20 - 305*n**18 + 37126*n**16 - 2348210*n**14 + 83969341*n**12 - 1729295165*n**10 + 20036895436*n**8 - 122579094320*n**6 + 354128908096*n**4 - 412509312000*n**2 + 162570240000))], [[2, 2, 2, 2, 2, 1], (-n**12 + 194*n**10 - 15285*n**8 + 501520*n**6 - 6974924*n**4 + 94322736*n**2 + 2034408960)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[4, 2, 2, 2, 1], (5*n**12 - 684*n**10 + 45753*n**8 - 1207274*n**6 - 4948008*n**4 - 261816192*n**2 + 261273600)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[3, 3, 2, 2, 1], 4*(n**12 - 127*n**10 + 9353*n**8 - 245981*n**6 - 6122934*n**4 + 50993928*n**2 - 32659200)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[6, 2, 2, 1], (-42*n**10 + 4164*n**8 - 204042*n**6 + 5352816*n**4 + 97818624*n**2 - 930579840)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[5, 3, 2, 1], -(28*n**6 + 515*n**4 + 108597*n**2 + 331740)/(n**2*(n**22 - 326*n**20 + 42511*n**18 - 2882036*n**16 + 111145711*n**14 - 2514183686*n**12 + 33325295041*n**10 - 253187957216*n**8 + 1061276784736*n**6 - 2307378836736*n**4 + 2291382432000*n**2 - 823011840000))], [[4, 4, 2, 1], (-25*n**8 - 710*n**6 - 99585*n**4 + 1884240*n**2 - 17645040)/(n**2*(n**24 - 335*n**22 + 45445*n**20 - 3264635*n**18 + 137084035*n**16 - 3514495085*n**14 + 55952948215*n**12 - 553115612585*n**10 + 3339968399680*n**8 - 11858869899360*n**6 + 23057791962624*n**4 - 21445453728000*n**2 + 7407106560000))], [[8, 2, 1], 143*(3*n**4 - 31*n**2 + 4108)/(n*(n**22 - 326*n**20 + 42511*n**18 - 2882036*n**16 + 111145711*n**14 - 2514183686*n**12 + 33325295041*n**10 - 253187957216*n**8 + 1061276784736*n**6 - 2307378836736*n**4 + 2291382432000*n**2 - 823011840000))], [[4, 3, 3, 1], (-20*n**10 + 676*n**8 - 165644*n**6 + 18713484*n**4 - 353590416*n**2 + 1005644160)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[7, 3, 1], 66*(4*n**10 + n**8 - 19082*n**6 + 51369*n**4 + 4886748*n**2 - 10886400)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[6, 4, 1], 6*(35*n**10 + 2438*n**8 - 441637*n**6 + 9646812*n**4 - 65050848*n**2 - 21772800)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[5, 5, 1], 28*(7*n**10 + 672*n**8 - 111237*n**6 + 2890118*n**4 - 30453480*n**2 + 60652800)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[10, 1], (-4862*n**2 + 318240)/(n**2*(n**20 - 385*n**18 + 61446*n**16 - 5293970*n**14 + 268880381*n**12 - 8261931405*n**10 + 151847872396*n**8 - 1593719752240*n**6 + 8689315795776*n**4 - 20407635072000*n**2 + 13168189440000))], [[3, 2, 2, 2, 2], 2*(n**12 - 138*n**10 + 10773*n**8 - 106744*n**6 + 10459356*n**4 + 334493712*n**2 - 2286144000)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[5, 2, 2, 2], (-14*n**10 + 1128*n**8 - 98694*n**6 - 871108*n**4 - 107204112*n**2 + 62933760)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[4, 3, 2, 2], (-10*n**8 + 470*n**6 - 79900*n**4 - 2159760*n**2 - 51315840)/(n**2*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[7, 2, 2], 44*(3*n**10 - 136*n**8 + 14063*n**6 + 285086*n**4 - 3323016*n**2 + 32659200)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[3, 3, 3, 2], (-8*n**10 + 400*n**8 - 89384*n**6 - 2455200*n**4 + 75327552*n**2 + 1140687360)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[6, 3, 2], 84*(n**10 + 17*n**8 + 5255*n**6 + 66403*n**4 - 3274236*n**2 + 1555200)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[5, 4, 2], 10*(7*n**8 + 477*n**6 + 38796*n**4 + 139136*n**2 + 17418240)/(n**3*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[9, 2], (-1430*n**4 + 20150*n**2 - 4343040)/(n**2*(n**22 - 386*n**20 + 61831*n**18 - 5355416*n**16 + 274174351*n**14 - 8530811786*n**12 + 160109803801*n**10 - 1745567624636*n**8 + 10283035548016*n**6 - 29096950867776*n**4 + 33575824512000*n**2 - 13168189440000))], [[5, 3, 3], 28*(2*n**10 + 107*n**8 + 31688*n**6 - 1068057*n**4 + 1813860*n**2 - 9331200)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[4, 4, 3], 10*(5*n**10 + 350*n**8 + 100205*n**6 - 4488384*n**4 + 47715696*n**2 + 169827840)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[8, 3], (-858*n**6 - 70590*n**4 - 1070472*n**2 + 74655360)/(n**2*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[7, 4], (-660*n**8 - 111720*n**6 + 3546060*n**4 - 25085520*n**2 - 494605440)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[6, 5], (-588*n**8 - 132888*n**6 + 5827668*n**4 - 131855472*n**2 + 1334309760)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[11], 16796/(n*(n**20 - 385*n**18 + 61446*n**16 - 5293970*n**14 + 268880381*n**12 - 8261931405*n**10 + 151847872396*n**8 - 1593719752240*n**6 + 8689315795776*n**4 - 20407635072000*n**2 + 13168189440000))]] -------------------------------------------------------------------------------- /MATHEMATICA/precomputedWG/functions11.txt: -------------------------------------------------------------------------------- 1 | [[[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], (n**18 - 344*n**16 + 47305*n**14 - 3338030*n**12 + 129218848*n**10 - 2728819832*n**8 + 29688278076*n**6 - 149061471624*n**4 + 273312649440*n**2 - 38799129600)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[2, 1, 1, 1, 1, 1, 1, 1, 1, 1], (-n**16 + 326*n**14 - 41797*n**12 + 2688500*n**10 - 91884112*n**8 + 1635300968*n**6 - 13995701820*n**4 + 49708341936*n**2 - 48846551040)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[3, 1, 1, 1, 1, 1, 1, 1, 1], 2*(n**16 - 306*n**14 + 36085*n**12 - 2074152*n**10 + 60751048*n**8 - 871809792*n**6 + 5519618316*n**4 - 12470338800*n**2 + 2155507200)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[2, 2, 1, 1, 1, 1, 1, 1, 1], (n**16 - 302*n**14 + 35119*n**12 - 1992224*n**10 + 57820486*n**8 - 828604784*n**6 + 5240619624*n**4 - 11643104160*n**2 + 6923750400)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[4, 1, 1, 1, 1, 1, 1, 1], (-5*n**14 + 1428*n**12 - 153587*n**10 + 7783266*n**8 - 190872794*n**6 + 2119020876*n**4 - 9309500784*n**2 + 11672760960)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[3, 2, 1, 1, 1, 1, 1, 1], (-2*n**12 + 528*n**10 - 50834*n**8 + 2184936*n**6 - 41042804*n**4 + 275906256*n**2 - 294877440)/(n**2*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[5, 1, 1, 1, 1, 1, 1], 14*(n**14 - 264*n**12 + 25567*n**10 - 1121958*n**8 + 22416562*n**6 - 183771348*n**4 + 508690800*n**2 - 158630400)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[2, 2, 2, 1, 1, 1, 1, 1], (-n**14 + 272*n**12 - 27939*n**10 + 1373134*n**8 - 33834566*n**6 + 392961564*n**4 - 1655003664*n**2 + 1788168960)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[4, 2, 1, 1, 1, 1, 1], 5*(n**12 - 233*n**10 + 18774*n**8 - 614998*n**6 + 7273496*n**4 - 19792128*n**2 + 34836480)/(n**3*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[3, 3, 1, 1, 1, 1, 1], 4*(n**14 - 229*n**12 + 18002*n**10 - 569693*n**8 + 6534687*n**6 - 22616928*n**4 + 38406960*n**2 + 97977600)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[6, 1, 1, 1, 1, 1], (-42*n**12 + 10068*n**10 - 857334*n**8 + 31554696*n**6 - 492637644*n**4 + 2842232976*n**2 - 4405415040)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[3, 2, 2, 1, 1, 1, 1], 2*(n**14 - 234*n**12 + 20167*n**10 - 830664*n**8 + 17947366*n**6 - 188054412*n**4 + 559562256*n**2 - 65318400)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[5, 2, 1, 1, 1, 1], (-14*n**8 + 1902*n**6 - 57796*n**4 + 304788*n**2 + 1139760)/(n**2*(n**22 - 326*n**20 + 42511*n**18 - 2882036*n**16 + 111145711*n**14 - 2514183686*n**12 + 33325295041*n**10 - 253187957216*n**8 + 1061276784736*n**6 - 2307378836736*n**4 + 2291382432000*n**2 - 823011840000))], [[4, 3, 1, 1, 1, 1], (-10*n**10 + 1702*n**8 - 71504*n**6 - 521212*n**4 + 50868624*n**2 - 167034240)/(n**2*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[7, 1, 1, 1, 1], 132*(n**12 - 211*n**10 + 15119*n**8 - 438257*n**6 + 4918740*n**4 - 18365472*n**2 + 10886400)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[2, 2, 2, 2, 1, 1, 1], (n**14 - 236*n**12 + 21033*n**10 - 889090*n**8 + 16846604*n**6 - 75657384*n**4 - 608866848*n**2 + 2482099200)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[4, 2, 2, 1, 1, 1], (-5*n**10 + 917*n**8 - 58276*n**6 + 1799668*n**4 - 31856304*n**2 + 168658560)/(n**2*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[3, 3, 2, 1, 1, 1], (-4*n**10 + 708*n**8 - 45506*n**6 + 1745352*n**4 - 47293830*n**2 + 468711360)/(n**2*(n**24 - 395*n**22 + 65305*n**20 - 5911895*n**18 + 322373095*n**16 - 10998380945*n**14 + 236887109875*n**12 - 3186555858845*n**10 + 25993144169740*n**8 - 121644270799920*n**6 + 295448382321984*n**4 - 315350610048000*n**2 + 118513704960000))], [[6, 2, 1, 1, 1], 6*(7*n**12 - 1207*n**10 + 64643*n**8 - 1240589*n**6 + 8910450*n**4 - 7330104*n**2 + 21772800)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[5, 3, 1, 1, 1], 7*(4*n**12 - 499*n**10 + 3908*n**8 + 1064929*n**6 - 27354462*n**4 + 155108520*n**2 - 177292800)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[4, 4, 1, 1, 1], (25*n**12 - 2620*n**10 - 66379*n**8 + 11938846*n**6 - 279926736*n**4 + 1516001184*n**2 - 653184000)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[8, 1, 1, 1], (-429*n**8 + 71643*n**6 - 3540966*n**4 + 51552072*n**2 - 84839040)/(n**2*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[3, 2, 2, 2, 1, 1], (-2*n**8 + 198*n**6 - 9012*n**4 + 58928*n**2 + 2211840)/(n**2*(n**22 - 309*n**20 + 38346*n**18 - 2496714*n**16 + 93362181*n**14 - 2065172529*n**12 + 26954076096*n**10 - 202726676064*n**8 + 844445285376*n**6 - 1829024944384*n**4 + 1812607488000*n**2 - 650280960000))], [[5, 2, 2, 1, 1], 2*(7*n**10 - 467*n**8 + 18200*n**6 - 327236*n**4 - 62960*n**2 + 5644800)/(n**3*(n**24 - 318*n**22 + 41127*n**20 - 2841828*n**18 + 115832607*n**16 - 2905432158*n**14 + 45540628857*n**12 - 445313360928*n**10 + 2668985369952*n**8 - 9429032512768*n**6 + 18273831987456*n**4 - 16963748352000*n**2 + 5852528640000))], [[4, 3, 2, 1, 1], 2*(5*n**8 - 153*n**6 + 12084*n**4 - 483008*n**2 + 537600)/(n**3*(n**22 - 309*n**20 + 38346*n**18 - 2496714*n**16 + 93362181*n**14 - 2065172529*n**12 + 26954076096*n**10 - 202726676064*n**8 + 844445285376*n**6 - 1829024944384*n**4 + 1812607488000*n**2 - 650280960000))], [[7, 2, 1, 1], (-132*n**6 + 6094*n**4 - 105754*n**2 + 1995840)/(n**2*(n**22 - 314*n**20 + 39871*n**18 - 2682344*n**16 + 105103231*n**14 - 2485019234*n**12 + 35600551921*n**10 - 302911153244*n**8 + 1457340756976*n**6 - 3599669484864*n**4 + 3875154048000*n**2 - 1463132160000))], [[3, 3, 3, 1, 1], 4*(2*n**8 - 46*n**6 + 10007*n**4 - 448380*n**2 + 806400)/(n**3*(n**22 - 309*n**20 + 38346*n**18 - 2496714*n**16 + 93362181*n**14 - 2065172529*n**12 + 26954076096*n**10 - 202726676064*n**8 + 844445285376*n**6 - 1829024944384*n**4 + 1812607488000*n**2 - 650280960000))], [[6, 3, 1, 1], (-84*n**6 - 2076*n**4 + 208848*n**2 - 140160)/(n**2*(n**22 - 309*n**20 + 38346*n**18 - 2496714*n**16 + 93362181*n**14 - 2065172529*n**12 + 26954076096*n**10 - 202726676064*n**8 + 844445285376*n**6 - 1829024944384*n**4 + 1812607488000*n**2 - 650280960000))], [[5, 4, 1, 1], (-70*n**8 - 4158*n**6 + 408324*n**4 - 5866448*n**2 + 16007040)/(n**2*(n**24 - 318*n**22 + 41127*n**20 - 2841828*n**18 + 115832607*n**16 - 2905432158*n**14 + 45540628857*n**12 - 445313360928*n**10 + 2668985369952*n**8 - 9429032512768*n**6 + 18273831987456*n**4 - 16963748352000*n**2 + 5852528640000))], [[9, 1, 1], 286*(5*n**2 - 194)/(n*(n**20 - 305*n**18 + 37126*n**16 - 2348210*n**14 + 83969341*n**12 - 1729295165*n**10 + 20036895436*n**8 - 122579094320*n**6 + 354128908096*n**4 - 412509312000*n**2 + 162570240000))], [[2, 2, 2, 2, 2, 1], (-n**12 + 194*n**10 - 15285*n**8 + 501520*n**6 - 6974924*n**4 + 94322736*n**2 + 2034408960)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[4, 2, 2, 2, 1], (5*n**12 - 684*n**10 + 45753*n**8 - 1207274*n**6 - 4948008*n**4 - 261816192*n**2 + 261273600)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[3, 3, 2, 2, 1], 4*(n**12 - 127*n**10 + 9353*n**8 - 245981*n**6 - 6122934*n**4 + 50993928*n**2 - 32659200)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[6, 2, 2, 1], (-42*n**10 + 4164*n**8 - 204042*n**6 + 5352816*n**4 + 97818624*n**2 - 930579840)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[5, 3, 2, 1], -(28*n**6 + 515*n**4 + 108597*n**2 + 331740)/(n**2*(n**22 - 326*n**20 + 42511*n**18 - 2882036*n**16 + 111145711*n**14 - 2514183686*n**12 + 33325295041*n**10 - 253187957216*n**8 + 1061276784736*n**6 - 2307378836736*n**4 + 2291382432000*n**2 - 823011840000))], [[4, 4, 2, 1], (-25*n**8 - 710*n**6 - 99585*n**4 + 1884240*n**2 - 17645040)/(n**2*(n**24 - 335*n**22 + 45445*n**20 - 3264635*n**18 + 137084035*n**16 - 3514495085*n**14 + 55952948215*n**12 - 553115612585*n**10 + 3339968399680*n**8 - 11858869899360*n**6 + 23057791962624*n**4 - 21445453728000*n**2 + 7407106560000))], [[8, 2, 1], 143*(3*n**4 - 31*n**2 + 4108)/(n*(n**22 - 326*n**20 + 42511*n**18 - 2882036*n**16 + 111145711*n**14 - 2514183686*n**12 + 33325295041*n**10 - 253187957216*n**8 + 1061276784736*n**6 - 2307378836736*n**4 + 2291382432000*n**2 - 823011840000))], [[4, 3, 3, 1], (-20*n**10 + 676*n**8 - 165644*n**6 + 18713484*n**4 - 353590416*n**2 + 1005644160)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[7, 3, 1], 66*(4*n**10 + n**8 - 19082*n**6 + 51369*n**4 + 4886748*n**2 - 10886400)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[6, 4, 1], 6*(35*n**10 + 2438*n**8 - 441637*n**6 + 9646812*n**4 - 65050848*n**2 - 21772800)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[5, 5, 1], 28*(7*n**10 + 672*n**8 - 111237*n**6 + 2890118*n**4 - 30453480*n**2 + 60652800)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[10, 1], (-4862*n**2 + 318240)/(n**2*(n**20 - 385*n**18 + 61446*n**16 - 5293970*n**14 + 268880381*n**12 - 8261931405*n**10 + 151847872396*n**8 - 1593719752240*n**6 + 8689315795776*n**4 - 20407635072000*n**2 + 13168189440000))], [[3, 2, 2, 2, 2], 2*(n**12 - 138*n**10 + 10773*n**8 - 106744*n**6 + 10459356*n**4 + 334493712*n**2 - 2286144000)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[5, 2, 2, 2], (-14*n**10 + 1128*n**8 - 98694*n**6 - 871108*n**4 - 107204112*n**2 + 62933760)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[4, 3, 2, 2], (-10*n**8 + 470*n**6 - 79900*n**4 - 2159760*n**2 - 51315840)/(n**2*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[7, 2, 2], 44*(3*n**10 - 136*n**8 + 14063*n**6 + 285086*n**4 - 3323016*n**2 + 32659200)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[3, 3, 3, 2], (-8*n**10 + 400*n**8 - 89384*n**6 - 2455200*n**4 + 75327552*n**2 + 1140687360)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[6, 3, 2], 84*(n**10 + 17*n**8 + 5255*n**6 + 66403*n**4 - 3274236*n**2 + 1555200)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[5, 4, 2], 10*(7*n**8 + 477*n**6 + 38796*n**4 + 139136*n**2 + 17418240)/(n**3*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[9, 2], (-1430*n**4 + 20150*n**2 - 4343040)/(n**2*(n**22 - 386*n**20 + 61831*n**18 - 5355416*n**16 + 274174351*n**14 - 8530811786*n**12 + 160109803801*n**10 - 1745567624636*n**8 + 10283035548016*n**6 - 29096950867776*n**4 + 33575824512000*n**2 - 13168189440000))], [[5, 3, 3], 28*(2*n**10 + 107*n**8 + 31688*n**6 - 1068057*n**4 + 1813860*n**2 - 9331200)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[4, 4, 3], 10*(5*n**10 + 350*n**8 + 100205*n**6 - 4488384*n**4 + 47715696*n**2 + 169827840)/(n**3*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[8, 3], (-858*n**6 - 70590*n**4 - 1070472*n**2 + 74655360)/(n**2*(n**24 - 390*n**22 + 63375*n**20 - 5602740*n**18 + 295596015*n**16 - 9627509190*n**14 + 194233050945*n**12 - 2386006839840*n**10 + 17265306046560*n**8 - 70229093059840*n**6 + 149963627983104*n**4 - 147471487488000*n**2 + 52672757760000))], [[7, 4], (-660*n**8 - 111720*n**6 + 3546060*n**4 - 25085520*n**2 - 494605440)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[6, 5], (-588*n**8 - 132888*n**6 + 5827668*n**4 - 131855472*n**2 + 1334309760)/(n**2*(n**26 - 399*n**24 + 66885*n**22 - 6173115*n**20 + 346020675*n**18 - 12287873325*n**16 + 280880633655*n**14 - 4134104298345*n**12 + 38739367605120*n**10 - 225616847478880*n**8 + 782025465521664*n**6 - 1497144139335936*n**4 + 1379916145152000*n**2 - 474054819840000))], [[11], 16796/(n*(n**20 - 385*n**18 + 61446*n**16 - 5293970*n**14 + 268880381*n**12 - 8261931405*n**10 + 151847872396*n**8 - 1593719752240*n**6 + 8689315795776*n**4 - 20407635072000*n**2 + 13168189440000))]] -------------------------------------------------------------------------------- /PYTHON/Weingarten/functions11.html: -------------------------------------------------------------------------------- 1 |

Weingarten functions for p = 11

Below are the values of the Weingarten function for permutation size p = 11. The input is given as a partition of p. You can also download them as a text file or as a python pickle file. 2 | 3 | \[\operatorname{Wg}([1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]) = \frac{n^{18} - 344 n^{16} + 47305 n^{14} - 3338030 n^{12} + 129218848 n^{10} - 2728819832 n^{8} + 29688278076 n^{6} - 149061471624 n^{4} + 273312649440 n^{2} - 38799129600}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 4 | \[\operatorname{Wg}([2, 1, 1, 1, 1, 1, 1, 1, 1, 1]) = \frac{- n^{16} + 326 n^{14} - 41797 n^{12} + 2688500 n^{10} - 91884112 n^{8} + 1635300968 n^{6} - 13995701820 n^{4} + 49708341936 n^{2} - 48846551040}{n^{2} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 5 | \[\operatorname{Wg}([3, 1, 1, 1, 1, 1, 1, 1, 1]) = \frac{2 n^{16} - 612 n^{14} + 72170 n^{12} - 4148304 n^{10} + 121502096 n^{8} - 1743619584 n^{6} + 11039236632 n^{4} - 24940677600 n^{2} + 4311014400}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 6 | \[\operatorname{Wg}([2, 2, 1, 1, 1, 1, 1, 1, 1]) = \frac{n^{16} - 302 n^{14} + 35119 n^{12} - 1992224 n^{10} + 57820486 n^{8} - 828604784 n^{6} + 5240619624 n^{4} - 11643104160 n^{2} + 6923750400}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 7 | \[\operatorname{Wg}([4, 1, 1, 1, 1, 1, 1, 1]) = \frac{- 5 n^{14} + 1428 n^{12} - 153587 n^{10} + 7783266 n^{8} - 190872794 n^{6} + 2119020876 n^{4} - 9309500784 n^{2} + 11672760960}{n^{2} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 8 | \[\operatorname{Wg}([3, 2, 1, 1, 1, 1, 1, 1]) = \frac{- 2 n^{12} + 528 n^{10} - 50834 n^{8} + 2184936 n^{6} - 41042804 n^{4} + 275906256 n^{2} - 294877440}{n^{2} \left(n^{24} - 390 n^{22} + 63375 n^{20} - 5602740 n^{18} + 295596015 n^{16} - 9627509190 n^{14} + 194233050945 n^{12} - 2386006839840 n^{10} + 17265306046560 n^{8} - 70229093059840 n^{6} + 149963627983104 n^{4} - 147471487488000 n^{2} + 52672757760000\right)}\] 9 | \[\operatorname{Wg}([5, 1, 1, 1, 1, 1, 1]) = \frac{14 n^{14} - 3696 n^{12} + 357938 n^{10} - 15707412 n^{8} + 313831868 n^{6} - 2572798872 n^{4} + 7121671200 n^{2} - 2220825600}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 10 | \[\operatorname{Wg}([2, 2, 2, 1, 1, 1, 1, 1]) = \frac{- n^{14} + 272 n^{12} - 27939 n^{10} + 1373134 n^{8} - 33834566 n^{6} + 392961564 n^{4} - 1655003664 n^{2} + 1788168960}{n^{2} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 11 | \[\operatorname{Wg}([4, 2, 1, 1, 1, 1, 1]) = \frac{5 n^{12} - 1165 n^{10} + 93870 n^{8} - 3074990 n^{6} + 36367480 n^{4} - 98960640 n^{2} + 174182400}{n^{3} \left(n^{24} - 390 n^{22} + 63375 n^{20} - 5602740 n^{18} + 295596015 n^{16} - 9627509190 n^{14} + 194233050945 n^{12} - 2386006839840 n^{10} + 17265306046560 n^{8} - 70229093059840 n^{6} + 149963627983104 n^{4} - 147471487488000 n^{2} + 52672757760000\right)}\] 12 | \[\operatorname{Wg}([3, 3, 1, 1, 1, 1, 1]) = \frac{4 n^{14} - 916 n^{12} + 72008 n^{10} - 2278772 n^{8} + 26138748 n^{6} - 90467712 n^{4} + 153627840 n^{2} + 391910400}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 13 | \[\operatorname{Wg}([6, 1, 1, 1, 1, 1]) = \frac{- 42 n^{12} + 10068 n^{10} - 857334 n^{8} + 31554696 n^{6} - 492637644 n^{4} + 2842232976 n^{2} - 4405415040}{n^{2} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 14 | \[\operatorname{Wg}([3, 2, 2, 1, 1, 1, 1]) = \frac{2 n^{14} - 468 n^{12} + 40334 n^{10} - 1661328 n^{8} + 35894732 n^{6} - 376108824 n^{4} + 1119124512 n^{2} - 130636800}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 15 | \[\operatorname{Wg}([5, 2, 1, 1, 1, 1]) = \frac{- 14 n^{8} + 1902 n^{6} - 57796 n^{4} + 304788 n^{2} + 1139760}{n^{2} \left(n^{22} - 326 n^{20} + 42511 n^{18} - 2882036 n^{16} + 111145711 n^{14} - 2514183686 n^{12} + 33325295041 n^{10} - 253187957216 n^{8} + 1061276784736 n^{6} - 2307378836736 n^{4} + 2291382432000 n^{2} - 823011840000\right)}\] 16 | \[\operatorname{Wg}([4, 3, 1, 1, 1, 1]) = \frac{- 10 n^{10} + 1702 n^{8} - 71504 n^{6} - 521212 n^{4} + 50868624 n^{2} - 167034240}{n^{2} \left(n^{24} - 390 n^{22} + 63375 n^{20} - 5602740 n^{18} + 295596015 n^{16} - 9627509190 n^{14} + 194233050945 n^{12} - 2386006839840 n^{10} + 17265306046560 n^{8} - 70229093059840 n^{6} + 149963627983104 n^{4} - 147471487488000 n^{2} + 52672757760000\right)}\] 17 | \[\operatorname{Wg}([7, 1, 1, 1, 1]) = \frac{132 n^{12} - 27852 n^{10} + 1995708 n^{8} - 57849924 n^{6} + 649273680 n^{4} - 2424242304 n^{2} + 1437004800}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 18 | \[\operatorname{Wg}([2, 2, 2, 2, 1, 1, 1]) = \frac{n^{14} - 236 n^{12} + 21033 n^{10} - 889090 n^{8} + 16846604 n^{6} - 75657384 n^{4} - 608866848 n^{2} + 2482099200}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 19 | \[\operatorname{Wg}([4, 2, 2, 1, 1, 1]) = \frac{- 5 n^{10} + 917 n^{8} - 58276 n^{6} + 1799668 n^{4} - 31856304 n^{2} + 168658560}{n^{2} \left(n^{24} - 390 n^{22} + 63375 n^{20} - 5602740 n^{18} + 295596015 n^{16} - 9627509190 n^{14} + 194233050945 n^{12} - 2386006839840 n^{10} + 17265306046560 n^{8} - 70229093059840 n^{6} + 149963627983104 n^{4} - 147471487488000 n^{2} + 52672757760000\right)}\] 20 | \[\operatorname{Wg}([3, 3, 2, 1, 1, 1]) = \frac{- 4 n^{10} + 708 n^{8} - 45506 n^{6} + 1745352 n^{4} - 47293830 n^{2} + 468711360}{n^{2} \left(n^{24} - 395 n^{22} + 65305 n^{20} - 5911895 n^{18} + 322373095 n^{16} - 10998380945 n^{14} + 236887109875 n^{12} - 3186555858845 n^{10} + 25993144169740 n^{8} - 121644270799920 n^{6} + 295448382321984 n^{4} - 315350610048000 n^{2} + 118513704960000\right)}\] 21 | \[\operatorname{Wg}([6, 2, 1, 1, 1]) = \frac{42 n^{12} - 7242 n^{10} + 387858 n^{8} - 7443534 n^{6} + 53462700 n^{4} - 43980624 n^{2} + 130636800}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 22 | \[\operatorname{Wg}([5, 3, 1, 1, 1]) = \frac{28 n^{12} - 3493 n^{10} + 27356 n^{8} + 7454503 n^{6} - 191481234 n^{4} + 1085759640 n^{2} - 1241049600}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 23 | \[\operatorname{Wg}([4, 4, 1, 1, 1]) = \frac{25 n^{12} - 2620 n^{10} - 66379 n^{8} + 11938846 n^{6} - 279926736 n^{4} + 1516001184 n^{2} - 653184000}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 24 | \[\operatorname{Wg}([8, 1, 1, 1]) = \frac{- 429 n^{8} + 71643 n^{6} - 3540966 n^{4} + 51552072 n^{2} - 84839040}{n^{2} \left(n^{24} - 390 n^{22} + 63375 n^{20} - 5602740 n^{18} + 295596015 n^{16} - 9627509190 n^{14} + 194233050945 n^{12} - 2386006839840 n^{10} + 17265306046560 n^{8} - 70229093059840 n^{6} + 149963627983104 n^{4} - 147471487488000 n^{2} + 52672757760000\right)}\] 25 | \[\operatorname{Wg}([3, 2, 2, 2, 1, 1]) = \frac{- 2 n^{8} + 198 n^{6} - 9012 n^{4} + 58928 n^{2} + 2211840}{n^{2} \left(n^{22} - 309 n^{20} + 38346 n^{18} - 2496714 n^{16} + 93362181 n^{14} - 2065172529 n^{12} + 26954076096 n^{10} - 202726676064 n^{8} + 844445285376 n^{6} - 1829024944384 n^{4} + 1812607488000 n^{2} - 650280960000\right)}\] 26 | \[\operatorname{Wg}([5, 2, 2, 1, 1]) = \frac{14 n^{10} - 934 n^{8} + 36400 n^{6} - 654472 n^{4} - 125920 n^{2} + 11289600}{n^{3} \left(n^{24} - 318 n^{22} + 41127 n^{20} - 2841828 n^{18} + 115832607 n^{16} - 2905432158 n^{14} + 45540628857 n^{12} - 445313360928 n^{10} + 2668985369952 n^{8} - 9429032512768 n^{6} + 18273831987456 n^{4} - 16963748352000 n^{2} + 5852528640000\right)}\] 27 | \[\operatorname{Wg}([4, 3, 2, 1, 1]) = \frac{10 n^{8} - 306 n^{6} + 24168 n^{4} - 966016 n^{2} + 1075200}{n^{3} \left(n^{22} - 309 n^{20} + 38346 n^{18} - 2496714 n^{16} + 93362181 n^{14} - 2065172529 n^{12} + 26954076096 n^{10} - 202726676064 n^{8} + 844445285376 n^{6} - 1829024944384 n^{4} + 1812607488000 n^{2} - 650280960000\right)}\] 28 | \[\operatorname{Wg}([7, 2, 1, 1]) = \frac{- 132 n^{6} + 6094 n^{4} - 105754 n^{2} + 1995840}{n^{2} \left(n^{22} - 314 n^{20} + 39871 n^{18} - 2682344 n^{16} + 105103231 n^{14} - 2485019234 n^{12} + 35600551921 n^{10} - 302911153244 n^{8} + 1457340756976 n^{6} - 3599669484864 n^{4} + 3875154048000 n^{2} - 1463132160000\right)}\] 29 | \[\operatorname{Wg}([3, 3, 3, 1, 1]) = \frac{8 n^{8} - 184 n^{6} + 40028 n^{4} - 1793520 n^{2} + 3225600}{n^{3} \left(n^{22} - 309 n^{20} + 38346 n^{18} - 2496714 n^{16} + 93362181 n^{14} - 2065172529 n^{12} + 26954076096 n^{10} - 202726676064 n^{8} + 844445285376 n^{6} - 1829024944384 n^{4} + 1812607488000 n^{2} - 650280960000\right)}\] 30 | \[\operatorname{Wg}([6, 3, 1, 1]) = \frac{- 84 n^{6} - 2076 n^{4} + 208848 n^{2} - 140160}{n^{2} \left(n^{22} - 309 n^{20} + 38346 n^{18} - 2496714 n^{16} + 93362181 n^{14} - 2065172529 n^{12} + 26954076096 n^{10} - 202726676064 n^{8} + 844445285376 n^{6} - 1829024944384 n^{4} + 1812607488000 n^{2} - 650280960000\right)}\] 31 | \[\operatorname{Wg}([5, 4, 1, 1]) = \frac{- 70 n^{8} - 4158 n^{6} + 408324 n^{4} - 5866448 n^{2} + 16007040}{n^{2} \left(n^{24} - 318 n^{22} + 41127 n^{20} - 2841828 n^{18} + 115832607 n^{16} - 2905432158 n^{14} + 45540628857 n^{12} - 445313360928 n^{10} + 2668985369952 n^{8} - 9429032512768 n^{6} + 18273831987456 n^{4} - 16963748352000 n^{2} + 5852528640000\right)}\] 32 | \[\operatorname{Wg}([9, 1, 1]) = \frac{1430 n^{2} - 55484}{n \left(n^{20} - 305 n^{18} + 37126 n^{16} - 2348210 n^{14} + 83969341 n^{12} - 1729295165 n^{10} + 20036895436 n^{8} - 122579094320 n^{6} + 354128908096 n^{4} - 412509312000 n^{2} + 162570240000\right)}\] 33 | \[\operatorname{Wg}([2, 2, 2, 2, 2, 1]) = \frac{- n^{12} + 194 n^{10} - 15285 n^{8} + 501520 n^{6} - 6974924 n^{4} + 94322736 n^{2} + 2034408960}{n^{2} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 34 | \[\operatorname{Wg}([4, 2, 2, 2, 1]) = \frac{5 n^{12} - 684 n^{10} + 45753 n^{8} - 1207274 n^{6} - 4948008 n^{4} - 261816192 n^{2} + 261273600}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 35 | \[\operatorname{Wg}([3, 3, 2, 2, 1]) = \frac{4 n^{12} - 508 n^{10} + 37412 n^{8} - 983924 n^{6} - 24491736 n^{4} + 203975712 n^{2} - 130636800}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 36 | \[\operatorname{Wg}([6, 2, 2, 1]) = \frac{- 42 n^{10} + 4164 n^{8} - 204042 n^{6} + 5352816 n^{4} + 97818624 n^{2} - 930579840}{n^{2} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 37 | \[\operatorname{Wg}([5, 3, 2, 1]) = \frac{- 28 n^{6} - 515 n^{4} - 108597 n^{2} - 331740}{n^{2} \left(n^{22} - 326 n^{20} + 42511 n^{18} - 2882036 n^{16} + 111145711 n^{14} - 2514183686 n^{12} + 33325295041 n^{10} - 253187957216 n^{8} + 1061276784736 n^{6} - 2307378836736 n^{4} + 2291382432000 n^{2} - 823011840000\right)}\] 38 | \[\operatorname{Wg}([4, 4, 2, 1]) = \frac{- 25 n^{8} - 710 n^{6} - 99585 n^{4} + 1884240 n^{2} - 17645040}{n^{2} \left(n^{24} - 335 n^{22} + 45445 n^{20} - 3264635 n^{18} + 137084035 n^{16} - 3514495085 n^{14} + 55952948215 n^{12} - 553115612585 n^{10} + 3339968399680 n^{8} - 11858869899360 n^{6} + 23057791962624 n^{4} - 21445453728000 n^{2} + 7407106560000\right)}\] 39 | \[\operatorname{Wg}([8, 2, 1]) = \frac{429 n^{4} - 4433 n^{2} + 587444}{n \left(n^{22} - 326 n^{20} + 42511 n^{18} - 2882036 n^{16} + 111145711 n^{14} - 2514183686 n^{12} + 33325295041 n^{10} - 253187957216 n^{8} + 1061276784736 n^{6} - 2307378836736 n^{4} + 2291382432000 n^{2} - 823011840000\right)}\] 40 | \[\operatorname{Wg}([4, 3, 3, 1]) = \frac{- 20 n^{10} + 676 n^{8} - 165644 n^{6} + 18713484 n^{4} - 353590416 n^{2} + 1005644160}{n^{2} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 41 | \[\operatorname{Wg}([7, 3, 1]) = \frac{264 n^{10} + 66 n^{8} - 1259412 n^{6} + 3390354 n^{4} + 322525368 n^{2} - 718502400}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 42 | \[\operatorname{Wg}([6, 4, 1]) = \frac{210 n^{10} + 14628 n^{8} - 2649822 n^{6} + 57880872 n^{4} - 390305088 n^{2} - 130636800}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 43 | \[\operatorname{Wg}([5, 5, 1]) = \frac{196 n^{10} + 18816 n^{8} - 3114636 n^{6} + 80923304 n^{4} - 852697440 n^{2} + 1698278400}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 44 | \[\operatorname{Wg}([10, 1]) = \frac{- 4862 n^{2} + 318240}{n^{2} \left(n^{20} - 385 n^{18} + 61446 n^{16} - 5293970 n^{14} + 268880381 n^{12} - 8261931405 n^{10} + 151847872396 n^{8} - 1593719752240 n^{6} + 8689315795776 n^{4} - 20407635072000 n^{2} + 13168189440000\right)}\] 45 | \[\operatorname{Wg}([3, 2, 2, 2, 2]) = \frac{2 n^{12} - 276 n^{10} + 21546 n^{8} - 213488 n^{6} + 20918712 n^{4} + 668987424 n^{2} - 4572288000}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 46 | \[\operatorname{Wg}([5, 2, 2, 2]) = \frac{- 14 n^{10} + 1128 n^{8} - 98694 n^{6} - 871108 n^{4} - 107204112 n^{2} + 62933760}{n^{2} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 47 | \[\operatorname{Wg}([4, 3, 2, 2]) = \frac{- 10 n^{8} + 470 n^{6} - 79900 n^{4} - 2159760 n^{2} - 51315840}{n^{2} \left(n^{24} - 390 n^{22} + 63375 n^{20} - 5602740 n^{18} + 295596015 n^{16} - 9627509190 n^{14} + 194233050945 n^{12} - 2386006839840 n^{10} + 17265306046560 n^{8} - 70229093059840 n^{6} + 149963627983104 n^{4} - 147471487488000 n^{2} + 52672757760000\right)}\] 48 | \[\operatorname{Wg}([7, 2, 2]) = \frac{132 n^{10} - 5984 n^{8} + 618772 n^{6} + 12543784 n^{4} - 146212704 n^{2} + 1437004800}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 49 | \[\operatorname{Wg}([3, 3, 3, 2]) = \frac{- 8 n^{10} + 400 n^{8} - 89384 n^{6} - 2455200 n^{4} + 75327552 n^{2} + 1140687360}{n^{2} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 50 | \[\operatorname{Wg}([6, 3, 2]) = \frac{84 n^{10} + 1428 n^{8} + 441420 n^{6} + 5577852 n^{4} - 275035824 n^{2} + 130636800}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 51 | \[\operatorname{Wg}([5, 4, 2]) = \frac{70 n^{8} + 4770 n^{6} + 387960 n^{4} + 1391360 n^{2} + 174182400}{n^{3} \left(n^{24} - 390 n^{22} + 63375 n^{20} - 5602740 n^{18} + 295596015 n^{16} - 9627509190 n^{14} + 194233050945 n^{12} - 2386006839840 n^{10} + 17265306046560 n^{8} - 70229093059840 n^{6} + 149963627983104 n^{4} - 147471487488000 n^{2} + 52672757760000\right)}\] 52 | \[\operatorname{Wg}([9, 2]) = \frac{- 1430 n^{4} + 20150 n^{2} - 4343040}{n^{2} \left(n^{22} - 386 n^{20} + 61831 n^{18} - 5355416 n^{16} + 274174351 n^{14} - 8530811786 n^{12} + 160109803801 n^{10} - 1745567624636 n^{8} + 10283035548016 n^{6} - 29096950867776 n^{4} + 33575824512000 n^{2} - 13168189440000\right)}\] 53 | \[\operatorname{Wg}([5, 3, 3]) = \frac{56 n^{10} + 2996 n^{8} + 887264 n^{6} - 29905596 n^{4} + 50788080 n^{2} - 261273600}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 54 | \[\operatorname{Wg}([4, 4, 3]) = \frac{50 n^{10} + 3500 n^{8} + 1002050 n^{6} - 44883840 n^{4} + 477156960 n^{2} + 1698278400}{n^{3} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 55 | \[\operatorname{Wg}([8, 3]) = \frac{- 858 n^{6} - 70590 n^{4} - 1070472 n^{2} + 74655360}{n^{2} \left(n^{24} - 390 n^{22} + 63375 n^{20} - 5602740 n^{18} + 295596015 n^{16} - 9627509190 n^{14} + 194233050945 n^{12} - 2386006839840 n^{10} + 17265306046560 n^{8} - 70229093059840 n^{6} + 149963627983104 n^{4} - 147471487488000 n^{2} + 52672757760000\right)}\] 56 | \[\operatorname{Wg}([7, 4]) = \frac{- 660 n^{8} - 111720 n^{6} + 3546060 n^{4} - 25085520 n^{2} - 494605440}{n^{2} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 57 | \[\operatorname{Wg}([6, 5]) = \frac{- 588 n^{8} - 132888 n^{6} + 5827668 n^{4} - 131855472 n^{2} + 1334309760}{n^{2} \left(n^{26} - 399 n^{24} + 66885 n^{22} - 6173115 n^{20} + 346020675 n^{18} - 12287873325 n^{16} + 280880633655 n^{14} - 4134104298345 n^{12} + 38739367605120 n^{10} - 225616847478880 n^{8} + 782025465521664 n^{6} - 1497144139335936 n^{4} + 1379916145152000 n^{2} - 474054819840000\right)}\] 58 | \[\operatorname{Wg}([11]) = \frac{16796}{n \left(n^{20} - 385 n^{18} + 61446 n^{16} - 5293970 n^{14} + 268880381 n^{12} - 8261931405 n^{10} + 151847872396 n^{8} - 1593719752240 n^{6} + 8689315795776 n^{4} - 20407635072000 n^{2} + 13168189440000\right)}\] 59 | --------------------------------------------------------------------------------