***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St000182 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of permutations whose cycle type is the given integer partition. This number is given by $$\{ \pi \in \mathfrak{S}_n : \text{type}(\pi) = \lambda\} = \frac{n!}{\lambda_1 \cdots \lambda_k \mu_1(\lambda)! \cdots \mu_n(\lambda)!}$$ where $\mu_j(\lambda)$ denotes the number of parts of $\lambda$ equal to $j$. All permutations with the same cycle type form a [[wikipedia:Conjugacy class]]. ----------------------------------------------------------------------------- References: [1] Section 1.3 p24 Kerber, A. Algebraic combinatorics via finite group actions [[MathSciNet:1115208]] ----------------------------------------------------------------------------- Code: def statistic(p): return p.conjugacy_class_size() ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 1 [1,1] => 1 [3] => 2 [2,1] => 3 [1,1,1] => 1 [4] => 6 [3,1] => 8 [2,2] => 3 [2,1,1] => 6 [1,1,1,1] => 1 [5] => 24 [4,1] => 30 [3,2] => 20 [3,1,1] => 20 [2,2,1] => 15 [2,1,1,1] => 10 [1,1,1,1,1] => 1 [6] => 120 [5,1] => 144 [4,2] => 90 [4,1,1] => 90 [3,3] => 40 [3,2,1] => 120 [3,1,1,1] => 40 [2,2,2] => 15 [2,2,1,1] => 45 [2,1,1,1,1] => 15 [1,1,1,1,1,1] => 1 [7] => 720 [6,1] => 840 [5,2] => 504 [5,1,1] => 504 [4,3] => 420 [4,2,1] => 630 [4,1,1,1] => 210 [3,3,1] => 280 [3,2,2] => 210 [3,2,1,1] => 420 [3,1,1,1,1] => 70 [2,2,2,1] => 105 [2,2,1,1,1] => 105 [2,1,1,1,1,1] => 21 [1,1,1,1,1,1,1] => 1 [8] => 5040 [7,1] => 5760 [6,2] => 3360 [6,1,1] => 3360 [5,3] => 2688 [5,2,1] => 4032 [5,1,1,1] => 1344 [4,4] => 1260 [4,3,1] => 3360 [4,2,2] => 1260 [4,2,1,1] => 2520 [4,1,1,1,1] => 420 [3,3,2] => 1120 [3,3,1,1] => 1120 [3,2,2,1] => 1680 [3,2,1,1,1] => 1120 [3,1,1,1,1,1] => 112 [2,2,2,2] => 105 [2,2,2,1,1] => 420 [2,2,1,1,1,1] => 210 [2,1,1,1,1,1,1] => 28 [1,1,1,1,1,1,1,1] => 1 [9] => 40320 [8,1] => 45360 [7,2] => 25920 [7,1,1] => 25920 [6,3] => 20160 [6,2,1] => 30240 [6,1,1,1] => 10080 [5,4] => 18144 [5,3,1] => 24192 [5,2,2] => 9072 [5,2,1,1] => 18144 [5,1,1,1,1] => 3024 [4,4,1] => 11340 [4,3,2] => 15120 [4,3,1,1] => 15120 [4,2,2,1] => 11340 [4,2,1,1,1] => 7560 [4,1,1,1,1,1] => 756 [3,3,3] => 2240 [3,3,2,1] => 10080 [3,3,1,1,1] => 3360 [3,2,2,2] => 2520 [3,2,2,1,1] => 7560 [3,2,1,1,1,1] => 2520 [3,1,1,1,1,1,1] => 168 [2,2,2,2,1] => 945 [2,2,2,1,1,1] => 1260 [2,2,1,1,1,1,1] => 378 [2,1,1,1,1,1,1,1] => 36 [1,1,1,1,1,1,1,1,1] => 1 [10] => 362880 [9,1] => 403200 [8,2] => 226800 [8,1,1] => 226800 [7,3] => 172800 [7,2,1] => 259200 [7,1,1,1] => 86400 [6,4] => 151200 [6,3,1] => 201600 [6,2,2] => 75600 [6,2,1,1] => 151200 [6,1,1,1,1] => 25200 [5,5] => 72576 [5,4,1] => 181440 [5,3,2] => 120960 [5,3,1,1] => 120960 [5,2,2,1] => 90720 [5,2,1,1,1] => 60480 [5,1,1,1,1,1] => 6048 [4,4,2] => 56700 [4,4,1,1] => 56700 [4,3,3] => 50400 [4,3,2,1] => 151200 [4,3,1,1,1] => 50400 [4,2,2,2] => 18900 [4,2,2,1,1] => 56700 [4,2,1,1,1,1] => 18900 [4,1,1,1,1,1,1] => 1260 [3,3,3,1] => 22400 [3,3,2,2] => 25200 [3,3,2,1,1] => 50400 [3,3,1,1,1,1] => 8400 [3,2,2,2,1] => 25200 [3,2,2,1,1,1] => 25200 [3,2,1,1,1,1,1] => 5040 [3,1,1,1,1,1,1,1] => 240 [2,2,2,2,2] => 945 [2,2,2,2,1,1] => 4725 [2,2,2,1,1,1,1] => 3150 [2,2,1,1,1,1,1,1] => 630 [2,1,1,1,1,1,1,1,1] => 45 [1,1,1,1,1,1,1,1,1,1] => 1 [11] => 3628800 [10,1] => 3991680 [9,1,1] => 2217600 [8,3] => 1663200 [7,4] => 1425600 [6,5] => 1330560 [6,4,1] => 1663200 [6,1,1,1,1,1] => 55440 [5,5,1] => 798336 [5,4,2] => 997920 [5,4,1,1] => 997920 [5,3,3] => 443520 [5,3,2,1] => 1330560 [5,3,1,1,1] => 443520 [5,2,2,2] => 166320 [5,2,2,1,1] => 498960 [5,2,1,1,1,1] => 166320 [4,4,3] => 415800 [4,4,2,1] => 623700 [4,4,1,1,1] => 207900 [4,3,3,1] => 554400 [4,3,2,2] => 415800 [4,3,2,1,1] => 831600 [4,2,2,2,1] => 207900 [3,3,3,2] => 123200 [3,3,3,1,1] => 123200 [3,3,2,2,1] => 277200 [3,2,2,2,2] => 34650 [2,2,2,2,2,1] => 10395 [2,1,1,1,1,1,1,1,1,1] => 55 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 39916800 [11,1] => 43545600 [10,1,1] => 23950080 [9,3] => 17740800 [7,5] => 13685760 [7,4,1] => 17107200 [6,6] => 6652800 [6,4,2] => 9979200 [5,5,2] => 4790016 [5,4,3] => 7983360 [5,4,2,1] => 11975040 [5,4,1,1,1] => 3991680 [5,3,3,1] => 5322240 [5,3,2,2] => 3991680 [5,3,2,1,1] => 7983360 [5,2,2,2,1] => 1995840 [5,2,2,1,1,1] => 1995840 [4,4,4] => 1247400 [4,4,3,1] => 4989600 [4,4,2,2] => 1871100 [4,4,2,1,1] => 3742200 [4,3,3,2] => 3326400 [4,3,3,1,1] => 3326400 [4,3,2,2,1] => 4989600 [3,3,3,3] => 246400 [3,3,3,2,1] => 1478400 [3,3,2,2,2] => 554400 [3,3,2,2,1,1] => 1663200 [3,2,2,2,2,1] => 415800 [2,2,2,2,2,2] => 10395 [1,1,1,1,1,1,1,1,1,1,1,1] => 1 [13] => 479001600 [12,1] => 518918400 [10,3] => 207567360 [8,5] => 155675520 [7,6] => 148262400 [7,5,1] => 177914880 [7,4,2] => 111196800 [6,6,1] => 86486400 [6,4,2,1] => 129729600 [5,5,3] => 41513472 [5,4,4] => 38918880 [5,4,3,1] => 103783680 [5,4,2,2] => 38918880 [5,4,2,1,1] => 77837760 [5,4,1,1,1,1] => 12972960 [5,3,3,2] => 34594560 [5,3,3,1,1] => 34594560 [5,3,2,2,1] => 51891840 [5,3,2,1,1,1] => 34594560 [4,4,4,1] => 16216200 [4,4,3,2] => 32432400 [4,4,3,1,1] => 32432400 [4,4,2,2,1] => 24324300 [4,3,3,3] => 9609600 [4,3,3,2,1] => 43243200 [3,3,3,3,1] => 3203200 [3,3,3,2,2] => 4804800 [3,3,2,2,2,1] => 7207200 [3,2,2,2,2,2] => 540540 [2,2,2,2,2,2,1] => 135135 [1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [9,5] => 1937295360 [7,7] => 889574400 [7,5,2] => 1245404160 [7,4,3] => 1037836800 [6,6,2] => 605404800 [6,4,4] => 454053600 [6,2,2,2,2] => 37837800 [5,5,4] => 435891456 [5,5,1,1,1,1] => 72648576 [5,4,3,2] => 726485760 [5,4,3,1,1] => 726485760 [5,4,2,2,1] => 544864320 [5,4,2,1,1,1] => 363242880 [5,3,3,3] => 107627520 [5,3,3,2,1] => 484323840 [5,2,2,2,2,1] => 45405360 [4,4,4,2] => 113513400 [4,4,3,3] => 151351200 [4,4,3,2,1] => 454053600 [4,3,2,2,2,1] => 151351200 [3,3,3,3,2] => 22422400 [3,3,3,3,1,1] => 22422400 [3,3,2,2,2,2] => 12612600 [2,2,2,2,2,2,2] => 135135 [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [6,5,1,1,1,1] => 1816214400 [6,3,3,3] => 1345344000 [6,2,2,2,2,1] => 567567000 [5,5,5] => 1743565824 [5,3,2,2,2,1] => 1816214400 [4,4,4,3] => 1135134000 [4,4,4,1,1,1] => 567567000 [4,3,3,3,2] => 1009008000 [3,3,3,3,3] => 44844800 [3,3,3,3,2,1] => 336336000 [3,3,3,2,2,2] => 168168000 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [3,3,3,3,2,2] => 1345344000 [2,2,2,2,2,2,2,2] => 2027025 [2,2,2,2,2,2,2,2,2] => 34459425 ----------------------------------------------------------------------------- Created: May 03, 2014 at 21:11 by Lahiru Kariyawasam ----------------------------------------------------------------------------- Last Updated: May 25, 2023 at 14:21 by Martin Rubey