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.
\[\operatorname{Wg}([1, 1, 1, 1, 1, 1, 1, 1, 1]) = \frac{n^{14} - 173 n^{12} + 11008 n^{10} - 317396 n^{8} + 4114204 n^{6} - 20599916 n^{4} + 29866032 n^{2} - 7902720}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([2, 1, 1, 1, 1, 1, 1, 1]) = \frac{- n^{10} + 155 n^{8} - 8372 n^{6} + 185110 n^{4} - 1441932 n^{2} + 1295280}{n^{2} \left(n^{18} - 205 n^{16} + 16626 n^{14} - 685610 n^{12} + 15408341 n^{10} - 188461065 n^{8} + 1190788936 n^{6} - 3500200720 n^{4} + 4108836096 n^{2} - 1625702400\right)}\]
\[\operatorname{Wg}([3, 1, 1, 1, 1, 1, 1]) = \frac{2 n^{12} - 288 n^{10} + 14114 n^{8} - 275976 n^{6} + 1933844 n^{4} - 3526416 n^{2} + 1128960}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([2, 2, 1, 1, 1, 1, 1]) = \frac{n^{12} - 139 n^{10} + 6552 n^{8} - 124098 n^{6} + 874052 n^{4} - 1498928 n^{2} - 376320}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([4, 1, 1, 1, 1, 1]) = \frac{- 5 n^{10} + 645 n^{8} - 26870 n^{6} + 405150 n^{4} - 1766600 n^{2} + 1538880}{n^{2} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([3, 2, 1, 1, 1, 1]) = \frac{- 2 n^{6} + 150 n^{4} - 1864 n^{2} + 3444}{n^{2} \left(n^{16} - 169 n^{14} + 10542 n^{12} - 306098 n^{10} + 4388813 n^{8} - 30463797 n^{6} + 94092244 n^{4} - 112879936 n^{2} + 45158400\right)}\]
\[\operatorname{Wg}([5, 1, 1, 1, 1]) = \frac{14 n^{10} - 1582 n^{8} + 54180 n^{6} - 595028 n^{4} + 1402576 n^{2} - 376320}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([2, 2, 2, 1, 1, 1]) = \frac{- n^{10} + 113 n^{8} - 4284 n^{6} + 69868 n^{4} - 440000 n^{2} - 321216}{n^{2} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([4, 2, 1, 1, 1]) = \frac{5 n^{8} - 427 n^{6} + 8350 n^{4} - 4568 n^{2} - 94080}{n^{3} \left(n^{18} - 205 n^{16} + 16626 n^{14} - 685610 n^{12} + 15408341 n^{10} - 188461065 n^{8} + 1190788936 n^{6} - 3500200720 n^{4} + 4108836096 n^{2} - 1625702400\right)}\]
\[\operatorname{Wg}([3, 3, 1, 1, 1]) = \frac{4 n^{6} - 288 n^{4} + 2306 n^{2} + 73578}{n \left(n^{18} - 205 n^{16} + 16626 n^{14} - 685610 n^{12} + 15408341 n^{10} - 188461065 n^{8} + 1190788936 n^{6} - 3500200720 n^{4} + 4108836096 n^{2} - 1625702400\right)}\]
\[\operatorname{Wg}([6, 1, 1, 1]) = \frac{- 42 n^{8} + 3966 n^{6} - 103620 n^{4} + 735744 n^{2} - 923328}{n^{2} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([3, 2, 2, 1, 1]) = \frac{2 n^{8} - 62 n^{6} + 1628 n^{4} - 14288 n^{2} + 7680}{n^{3} \left(n^{18} - 160 n^{16} + 9606 n^{14} - 281420 n^{12} + 4361201 n^{10} - 36395580 n^{8} + 161249776 n^{6} - 362117440 n^{4} + 365884416 n^{2} - 132710400\right)}\]
\[\operatorname{Wg}([5, 2, 1, 1]) = \frac{- 14 n^{4} + 124 n^{2} + 1360}{n^{2} \left(n^{16} - 156 n^{14} + 8982 n^{12} - 245492 n^{10} + 3379233 n^{8} - 22878648 n^{6} + 69735184 n^{4} - 83176704 n^{2} + 33177600\right)}\]
\[\operatorname{Wg}([4, 3, 1, 1]) = \frac{- 10 n^{6} - 174 n^{4} + 7240 n^{2} - 12096}{n^{2} \left(n^{18} - 160 n^{16} + 9606 n^{14} - 281420 n^{12} + 4361201 n^{10} - 36395580 n^{8} + 161249776 n^{6} - 362117440 n^{4} + 365884416 n^{2} - 132710400\right)}\]
\[\operatorname{Wg}([7, 1, 1]) = \frac{132 n^{2} - 2442}{n \left(n^{16} - 156 n^{14} + 8982 n^{12} - 245492 n^{10} + 3379233 n^{8} - 22878648 n^{6} + 69735184 n^{4} - 83176704 n^{2} + 33177600\right)}\]
\[\operatorname{Wg}([2, 2, 2, 2, 1]) = \frac{n^{10} - 81 n^{8} + 2892 n^{6} - 22924 n^{4} - 534288 n^{2} + 1128960}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([4, 2, 2, 1]) = \frac{- 5 n^{8} + 219 n^{6} - 6926 n^{4} + 181656 n^{2} - 23744}{n^{2} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([3, 3, 2, 1]) = \frac{- 4 n^{4} - 32 n^{2} - 7308}{n^{2} \left(n^{16} - 169 n^{14} + 10542 n^{12} - 306098 n^{10} + 4388813 n^{8} - 30463797 n^{6} + 94092244 n^{4} - 112879936 n^{2} + 45158400\right)}\]
\[\operatorname{Wg}([6, 2, 1]) = \frac{42 n^{4} + 204 n^{2} + 7962}{n \left(n^{18} - 173 n^{16} + 11218 n^{14} - 348266 n^{12} + 5613205 n^{10} - 48019049 n^{8} + 215947432 n^{6} - 489248912 n^{4} + 496678144 n^{2} - 180633600\right)}\]
\[\operatorname{Wg}([5, 3, 1]) = \frac{28 n^{8} + 483 n^{6} - 66843 n^{4} + 362012 n^{2} + 188160}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([4, 4, 1]) = \frac{25 n^{8} + 935 n^{6} - 89200 n^{4} + 736720 n^{2} - 376320}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([8, 1]) = \frac{- 429 n^{2} + 16016}{n^{2} \left(n^{16} - 204 n^{14} + 16422 n^{12} - 669188 n^{10} + 14739153 n^{8} - 173721912 n^{6} + 1017067024 n^{4} - 2483133696 n^{2} + 1625702400\right)}\]
\[\operatorname{Wg}([3, 2, 2, 2]) = \frac{- 2 n^{8} + 78 n^{6} - 5388 n^{4} - 115648 n^{2} - 665280}{n^{2} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([5, 2, 2]) = \frac{14 n^{8} - 66 n^{6} + 26916 n^{4} + 288976 n^{2} - 376320}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([4, 3, 2]) = \frac{10 n^{6} + 270 n^{4} + 33320 n^{2} - 94080}{n^{3} \left(n^{18} - 205 n^{16} + 16626 n^{14} - 685610 n^{12} + 15408341 n^{10} - 188461065 n^{8} + 1190788936 n^{6} - 3500200720 n^{4} + 4108836096 n^{2} - 1625702400\right)}\]
\[\operatorname{Wg}([7, 2]) = \frac{- 132 n^{4} - 1100 n^{2} - 109648}{n^{2} \left(n^{18} - 205 n^{16} + 16626 n^{14} - 685610 n^{12} + 15408341 n^{10} - 188461065 n^{8} + 1190788936 n^{6} - 3500200720 n^{4} + 4108836096 n^{2} - 1625702400\right)}\]
\[\operatorname{Wg}([3, 3, 3]) = \frac{8 n^{8} + 248 n^{6} + 48992 n^{4} - 825408 n^{2} - 1128960}{n^{3} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([6, 3]) = \frac{- 84 n^{6} - 5964 n^{4} + 41664 n^{2} + 538944}{n^{2} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([5, 4]) = \frac{- 70 n^{6} - 8050 n^{4} + 142520 n^{2} - 1041600}{n^{2} \left(n^{20} - 209 n^{18} + 17446 n^{16} - 752114 n^{14} + 18150781 n^{12} - 250094429 n^{10} + 1944633196 n^{8} - 8263356464 n^{6} + 18109638976 n^{4} - 18061046784 n^{2} + 6502809600\right)}\]
\[\operatorname{Wg}([9]) = \frac{1430}{n \left(n^{16} - 204 n^{14} + 16422 n^{12} - 669188 n^{10} + 14739153 n^{8} - 173721912 n^{6} + 1017067024 n^{4} - 2483133696 n^{2} + 1625702400\right)}\]