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.
\[\operatorname{Wg}([1, 1, 1, 1, 1, 1, 1, 1]) = \frac{n^{12} - 117 n^{10} + 4792 n^{8} - 82644 n^{6} + 573772 n^{4} - 1337484 n^{2} + 771120}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\]
\[\operatorname{Wg}([2, 1, 1, 1, 1, 1, 1]) = \frac{- n^{10} + 105 n^{8} - 3682 n^{6} + 50490 n^{4} - 247552 n^{2} + 331680}{n \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\]
\[\operatorname{Wg}([3, 1, 1, 1, 1, 1]) = \frac{2 n^{10} - 185 n^{8} + 5359 n^{6} - 54370 n^{4} + 167634 n^{2} - 128520}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\]
\[\operatorname{Wg}([2, 2, 1, 1, 1, 1]) = \frac{n^{10} - 87 n^{8} + 2356 n^{6} - 23058 n^{4} + 76228 n^{2} - 5040}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\]
\[\operatorname{Wg}([4, 1, 1, 1, 1]) = \frac{- 5 n^{8} + 401 n^{6} - 9322 n^{4} + 65494 n^{2} - 106968}{n \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\]
\[\operatorname{Wg}([3, 2, 1, 1, 1]) = \frac{- 2 n^{8} + 131 n^{6} - 2197 n^{4} + 10084 n^{2} - 28176}{n \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\]
\[\operatorname{Wg}([5, 1, 1, 1]) = \frac{14 n^{8} - 938 n^{6} + 16576 n^{4} - 76132 n^{2} + 65520}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\]
\[\operatorname{Wg}([2, 2, 2, 1, 1]) = \frac{- n^{4} + 23 n^{2} - 310}{n \left(n^{14} - 105 n^{12} + 3822 n^{10} - 61490 n^{8} + 453453 n^{6} - 1442805 n^{4} + 1752724 n^{2} - 705600\right)}\]
\[\operatorname{Wg}([4, 2, 1, 1]) = \frac{5 n^{4} - 13 n^{2} - 280}{n^{2} \left(n^{14} - 105 n^{12} + 3822 n^{10} - 61490 n^{8} + 453453 n^{6} - 1442805 n^{4} + 1752724 n^{2} - 705600\right)}\]
\[\operatorname{Wg}([3, 3, 1, 1]) = \frac{4 n^{6} + 28 n^{4} - 806 n^{2} + 630}{n^{2} \left(n^{16} - 109 n^{14} + 4242 n^{12} - 76778 n^{10} + 699413 n^{8} - 3256617 n^{6} + 7523944 n^{4} - 7716496 n^{2} + 2822400\right)}\]
\[\operatorname{Wg}([6, 1, 1]) = \frac{- 42 n^{2} + 474}{n \left(n^{14} - 105 n^{12} + 3822 n^{10} - 61490 n^{8} + 453453 n^{6} - 1442805 n^{4} + 1752724 n^{2} - 705600\right)}\]
\[\operatorname{Wg}([3, 2, 2, 1]) = \frac{2 n^{8} - 65 n^{6} + 1753 n^{4} - 34450 n^{2} + 83160}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\]
\[\operatorname{Wg}([5, 2, 1]) = \frac{- 14 n^{4} - 100 n^{2} - 96}{n \left(n^{16} - 120 n^{14} + 5166 n^{12} - 100340 n^{10} + 954921 n^{8} - 4562460 n^{6} + 10700656 n^{4} - 11062080 n^{2} + 4064256\right)}\]
\[\operatorname{Wg}([4, 3, 1]) = \frac{- 10 n^{6} - 127 n^{4} + 12605 n^{2} - 62868}{n \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\]
\[\operatorname{Wg}([7, 1]) = \frac{132 n^{2} - 3465}{n^{2} \left(n^{14} - 140 n^{12} + 7462 n^{10} - 191620 n^{8} + 2475473 n^{6} - 15291640 n^{4} + 38402064 n^{2} - 25401600\right)}\]
\[\operatorname{Wg}([2, 2, 2, 2]) = \frac{n^{8} - 33 n^{6} + 1404 n^{4} + 23828 n^{2} + 65520}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\]
\[\operatorname{Wg}([4, 2, 2]) = \frac{- 5 n^{6} - 5 n^{4} - 6590 n^{2} - 3480}{n \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\]
\[\operatorname{Wg}([3, 3, 2]) = \frac{- 4 n^{6} - 52 n^{4} - 8056 n^{2} + 63552}{n \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\]
\[\operatorname{Wg}([6, 2]) = \frac{42 n^{4} + 588 n^{2} + 14490}{n^{2} \left(n^{16} - 141 n^{14} + 7602 n^{12} - 199082 n^{10} + 2667093 n^{8} - 17767113 n^{6} + 53693704 n^{4} - 63803664 n^{2} + 25401600\right)}\]
\[\operatorname{Wg}([5, 3]) = \frac{28 n^{6} + 1645 n^{4} - 15533 n^{2} + 3780}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\]
\[\operatorname{Wg}([4, 4]) = \frac{25 n^{6} + 1975 n^{4} - 27200 n^{2} + 136080}{n^{2} \left(n^{18} - 145 n^{16} + 8166 n^{14} - 229490 n^{12} + 3463421 n^{10} - 28435485 n^{8} + 124762156 n^{6} - 278578480 n^{4} + 280616256 n^{2} - 101606400\right)}\]
\[\operatorname{Wg}([8]) = - \frac{429}{n \left(n^{14} - 140 n^{12} + 7462 n^{10} - 191620 n^{8} + 2475473 n^{6} - 15291640 n^{4} + 38402064 n^{2} - 25401600\right)}\]