***************************************************************************** * 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: St000048 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The multinomial of the parts of a partition. Given an integer partition $\lambda = [\lambda_1,\ldots,\lambda_k]$, this is the multinomial $$\binom{|\lambda|}{\lambda_1,\ldots,\lambda_k}.$$ For any integer composition $\mu$ that is a rearrangement of $\lambda$, this is the number of ordered set partitions whose list of block sizes is $\mu$. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(L): return multinomial(list(L)) ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 1 [1,1] => 2 [3] => 1 [2,1] => 3 [1,1,1] => 6 [4] => 1 [3,1] => 4 [2,2] => 6 [2,1,1] => 12 [1,1,1,1] => 24 [5] => 1 [4,1] => 5 [3,2] => 10 [3,1,1] => 20 [2,2,1] => 30 [2,1,1,1] => 60 [1,1,1,1,1] => 120 [6] => 1 [5,1] => 6 [4,2] => 15 [4,1,1] => 30 [3,3] => 20 [3,2,1] => 60 [3,1,1,1] => 120 [2,2,2] => 90 [2,2,1,1] => 180 [2,1,1,1,1] => 360 [1,1,1,1,1,1] => 720 [7] => 1 [6,1] => 7 [5,2] => 21 [5,1,1] => 42 [4,3] => 35 [4,2,1] => 105 [4,1,1,1] => 210 [3,3,1] => 140 [3,2,2] => 210 [3,2,1,1] => 420 [3,1,1,1,1] => 840 [2,2,2,1] => 630 [2,2,1,1,1] => 1260 [2,1,1,1,1,1] => 2520 [1,1,1,1,1,1,1] => 5040 [8] => 1 [7,1] => 8 [6,2] => 28 [6,1,1] => 56 [5,3] => 56 [5,2,1] => 168 [5,1,1,1] => 336 [4,4] => 70 [4,3,1] => 280 [4,2,2] => 420 [4,2,1,1] => 840 [4,1,1,1,1] => 1680 [3,3,2] => 560 [3,3,1,1] => 1120 [3,2,2,1] => 1680 [3,2,1,1,1] => 3360 [3,1,1,1,1,1] => 6720 [2,2,2,2] => 2520 [2,2,2,1,1] => 5040 [2,2,1,1,1,1] => 10080 [2,1,1,1,1,1,1] => 20160 [1,1,1,1,1,1,1,1] => 40320 [9] => 1 [8,1] => 9 [7,2] => 36 [7,1,1] => 72 [6,3] => 84 [6,2,1] => 252 [6,1,1,1] => 504 [5,4] => 126 [5,3,1] => 504 [5,2,2] => 756 [5,2,1,1] => 1512 [5,1,1,1,1] => 3024 [4,4,1] => 630 [4,3,2] => 1260 [4,3,1,1] => 2520 [4,2,2,1] => 3780 [4,2,1,1,1] => 7560 [4,1,1,1,1,1] => 15120 [3,3,3] => 1680 [3,3,2,1] => 5040 [3,3,1,1,1] => 10080 [3,2,2,2] => 7560 [3,2,2,1,1] => 15120 [3,2,1,1,1,1] => 30240 [3,1,1,1,1,1,1] => 60480 [2,2,2,2,1] => 22680 [2,2,2,1,1,1] => 45360 [2,2,1,1,1,1,1] => 90720 [2,1,1,1,1,1,1,1] => 181440 [1,1,1,1,1,1,1,1,1] => 362880 [10] => 1 [9,1] => 10 [8,2] => 45 [8,1,1] => 90 [7,3] => 120 [7,2,1] => 360 [7,1,1,1] => 720 [6,4] => 210 [6,3,1] => 840 [6,2,2] => 1260 [6,2,1,1] => 2520 [6,1,1,1,1] => 5040 [5,5] => 252 [5,4,1] => 1260 [5,3,2] => 2520 [5,3,1,1] => 5040 [5,2,2,1] => 7560 [5,2,1,1,1] => 15120 [5,1,1,1,1,1] => 30240 [4,4,2] => 3150 [4,4,1,1] => 6300 [4,3,3] => 4200 [4,3,2,1] => 12600 [4,3,1,1,1] => 25200 [4,2,2,2] => 18900 [4,2,2,1,1] => 37800 [4,2,1,1,1,1] => 75600 [4,1,1,1,1,1,1] => 151200 [3,3,3,1] => 16800 [3,3,2,2] => 25200 [3,3,2,1,1] => 50400 [3,3,1,1,1,1] => 100800 [3,2,2,2,1] => 75600 [3,2,2,1,1,1] => 151200 [3,2,1,1,1,1,1] => 302400 [3,1,1,1,1,1,1,1] => 604800 [2,2,2,2,2] => 113400 [2,2,2,2,1,1] => 226800 [2,2,2,1,1,1,1] => 453600 [2,2,1,1,1,1,1,1] => 907200 [2,1,1,1,1,1,1,1,1] => 1814400 [1,1,1,1,1,1,1,1,1,1] => 3628800 [11] => 1 [10,1] => 11 [9,2] => 55 [9,1,1] => 110 [8,3] => 165 [8,2,1] => 495 [8,1,1,1] => 990 [7,4] => 330 [7,3,1] => 1320 [7,2,2] => 1980 [7,2,1,1] => 3960 [7,1,1,1,1] => 7920 [6,5] => 462 [6,4,1] => 2310 [6,3,2] => 4620 [6,3,1,1] => 9240 [6,2,2,1] => 13860 [6,2,1,1,1] => 27720 [6,1,1,1,1,1] => 55440 [5,5,1] => 2772 [5,4,2] => 6930 [5,4,1,1] => 13860 [5,3,3] => 9240 [5,3,2,1] => 27720 [5,3,1,1,1] => 55440 [5,2,2,2] => 41580 [5,2,2,1,1] => 83160 [5,2,1,1,1,1] => 166320 [5,1,1,1,1,1,1] => 332640 [4,4,3] => 11550 [4,4,2,1] => 34650 [4,4,1,1,1] => 69300 [4,3,3,1] => 46200 [4,3,2,2] => 69300 [4,3,2,1,1] => 138600 [4,3,1,1,1,1] => 277200 [4,2,2,2,1] => 207900 [4,2,2,1,1,1] => 415800 [4,2,1,1,1,1,1] => 831600 [4,1,1,1,1,1,1,1] => 1663200 [3,3,3,2] => 92400 [3,3,3,1,1] => 184800 [3,3,2,2,1] => 277200 [3,3,2,1,1,1] => 554400 [3,3,1,1,1,1,1] => 1108800 [3,2,2,2,2] => 415800 [3,2,2,2,1,1] => 831600 [3,2,2,1,1,1,1] => 1663200 [3,2,1,1,1,1,1,1] => 3326400 [3,1,1,1,1,1,1,1,1] => 6652800 [2,2,2,2,2,1] => 1247400 [2,2,2,2,1,1,1] => 2494800 [2,2,2,1,1,1,1,1] => 4989600 [2,2,1,1,1,1,1,1,1] => 9979200 [12] => 1 [11,1] => 12 [10,2] => 66 [10,1,1] => 132 [9,3] => 220 [9,2,1] => 660 [9,1,1,1] => 1320 [8,4] => 495 [8,3,1] => 1980 [8,2,2] => 2970 [8,2,1,1] => 5940 [8,1,1,1,1] => 11880 [7,5] => 792 [7,4,1] => 3960 [7,3,2] => 7920 [7,3,1,1] => 15840 [7,2,2,1] => 23760 [7,2,1,1,1] => 47520 [7,1,1,1,1,1] => 95040 [6,6] => 924 [6,5,1] => 5544 [6,4,2] => 13860 [6,4,1,1] => 27720 [6,3,3] => 18480 [6,3,2,1] => 55440 [6,3,1,1,1] => 110880 [6,2,2,2] => 83160 [6,2,2,1,1] => 166320 [6,2,1,1,1,1] => 332640 [6,1,1,1,1,1,1] => 665280 [5,5,2] => 16632 [5,5,1,1] => 33264 [5,4,3] => 27720 [5,4,2,1] => 83160 [5,4,1,1,1] => 166320 [5,3,3,1] => 110880 [5,3,2,2] => 166320 [5,3,2,1,1] => 332640 [5,3,1,1,1,1] => 665280 [5,2,2,2,1] => 498960 [5,2,2,1,1,1] => 997920 [5,2,1,1,1,1,1] => 1995840 [5,1,1,1,1,1,1,1] => 3991680 [4,4,4] => 34650 [4,4,3,1] => 138600 [4,4,2,2] => 207900 [4,4,2,1,1] => 415800 [4,4,1,1,1,1] => 831600 [4,3,3,2] => 277200 [4,3,3,1,1] => 554400 [4,3,2,2,1] => 831600 [4,3,2,1,1,1] => 1663200 [4,3,1,1,1,1,1] => 3326400 [4,2,2,2,2] => 1247400 [4,2,2,2,1,1] => 2494800 [4,2,2,1,1,1,1] => 4989600 [4,2,1,1,1,1,1,1] => 9979200 [3,3,3,3] => 369600 [3,3,3,2,1] => 1108800 [3,3,3,1,1,1] => 2217600 [3,3,2,2,2] => 1663200 [3,3,2,2,1,1] => 3326400 [3,3,2,1,1,1,1] => 6652800 [3,2,2,2,2,1] => 4989600 [3,2,2,2,1,1,1] => 9979200 [2,2,2,2,2,2] => 7484400 [13] => 1 [12,1] => 13 [11,2] => 78 [11,1,1] => 156 [10,3] => 286 [10,2,1] => 858 [10,1,1,1] => 1716 [9,4] => 715 [9,3,1] => 2860 [9,2,2] => 4290 [9,2,1,1] => 8580 [9,1,1,1,1] => 17160 [8,5] => 1287 [8,4,1] => 6435 [8,3,2] => 12870 [8,3,1,1] => 25740 [8,2,2,1] => 38610 [8,2,1,1,1] => 77220 [8,1,1,1,1,1] => 154440 [7,6] => 1716 [7,5,1] => 10296 [7,4,2] => 25740 [7,4,1,1] => 51480 [7,3,3] => 34320 [7,3,2,1] => 102960 [7,3,1,1,1] => 205920 [7,2,2,2] => 154440 [7,2,2,1,1] => 308880 [7,2,1,1,1,1] => 617760 [7,1,1,1,1,1,1] => 1235520 [6,6,1] => 12012 [6,5,2] => 36036 [6,5,1,1] => 72072 [6,4,3] => 60060 [6,4,2,1] => 180180 [6,4,1,1,1] => 360360 [6,3,3,1] => 240240 [6,3,2,2] => 360360 [6,3,2,1,1] => 720720 [6,3,1,1,1,1] => 1441440 [6,2,2,2,1] => 1081080 [6,2,2,1,1,1] => 2162160 [6,2,1,1,1,1,1] => 4324320 [6,1,1,1,1,1,1,1] => 8648640 [5,5,3] => 72072 [5,5,2,1] => 216216 [5,5,1,1,1] => 432432 [5,4,4] => 90090 [5,4,3,1] => 360360 [5,4,2,2] => 540540 [5,4,2,1,1] => 1081080 [5,4,1,1,1,1] => 2162160 [5,3,3,2] => 720720 [5,3,3,1,1] => 1441440 [5,3,2,2,1] => 2162160 [5,3,2,1,1,1] => 4324320 [5,3,1,1,1,1,1] => 8648640 [5,2,2,2,2] => 3243240 [5,2,2,2,1,1] => 6486480 [4,4,4,1] => 450450 [4,4,3,2] => 900900 [4,4,3,1,1] => 1801800 [4,4,2,2,1] => 2702700 [4,4,2,1,1,1] => 5405400 [4,3,3,3] => 1201200 [4,3,3,2,1] => 3603600 [4,3,3,1,1,1] => 7207200 [4,3,2,2,2] => 5405400 [3,3,3,3,1] => 4804800 [3,3,3,2,2] => 7207200 [14] => 1 [13,1] => 14 [12,2] => 91 [12,1,1] => 182 [11,3] => 364 [11,2,1] => 1092 [11,1,1,1] => 2184 [10,4] => 1001 [10,3,1] => 4004 [10,2,2] => 6006 [10,2,1,1] => 12012 [10,1,1,1,1] => 24024 [9,5] => 2002 [9,4,1] => 10010 [9,3,2] => 20020 [9,3,1,1] => 40040 [9,2,2,1] => 60060 [9,2,1,1,1] => 120120 [9,1,1,1,1,1] => 240240 [8,6] => 3003 [8,5,1] => 18018 [8,4,2] => 45045 [8,4,1,1] => 90090 [8,3,3] => 60060 [8,3,2,1] => 180180 [8,3,1,1,1] => 360360 [8,2,2,2] => 270270 [8,2,2,1,1] => 540540 [8,2,1,1,1,1] => 1081080 [8,1,1,1,1,1,1] => 2162160 [7,7] => 3432 [7,6,1] => 24024 [7,5,2] => 72072 [7,5,1,1] => 144144 [7,4,3] => 120120 [7,4,2,1] => 360360 [7,4,1,1,1] => 720720 [7,3,3,1] => 480480 [7,3,2,2] => 720720 [7,3,2,1,1] => 1441440 [7,3,1,1,1,1] => 2882880 [7,2,2,2,1] => 2162160 [7,2,2,1,1,1] => 4324320 [7,2,1,1,1,1,1] => 8648640 [6,6,2] => 84084 [6,6,1,1] => 168168 [6,5,3] => 168168 [6,5,2,1] => 504504 [6,5,1,1,1] => 1009008 [6,4,4] => 210210 [6,4,3,1] => 840840 [6,4,2,2] => 1261260 [6,4,2,1,1] => 2522520 [6,4,1,1,1,1] => 5045040 [6,3,3,2] => 1681680 [6,3,3,1,1] => 3363360 [6,3,2,2,1] => 5045040 [6,2,2,2,2] => 7567560 [5,5,4] => 252252 [5,5,3,1] => 1009008 [5,5,2,2] => 1513512 [5,5,2,1,1] => 3027024 [5,5,1,1,1,1] => 6054048 [5,4,4,1] => 1261260 [5,4,3,2] => 2522520 [5,4,3,1,1] => 5045040 [5,4,2,2,1] => 7567560 [5,3,3,3] => 3363360 [4,4,4,2] => 3153150 [4,4,4,1,1] => 6306300 [4,4,3,3] => 4204200 [15] => 1 [14,1] => 15 [13,2] => 105 [13,1,1] => 210 [12,3] => 455 [12,2,1] => 1365 [12,1,1,1] => 2730 [11,4] => 1365 [11,3,1] => 5460 [11,2,2] => 8190 [11,2,1,1] => 16380 [11,1,1,1,1] => 32760 [10,5] => 3003 [10,4,1] => 15015 [10,3,2] => 30030 [10,3,1,1] => 60060 [10,2,2,1] => 90090 [10,2,1,1,1] => 180180 [10,1,1,1,1,1] => 360360 [9,6] => 5005 [9,5,1] => 30030 [9,4,2] => 75075 [9,4,1,1] => 150150 [9,3,3] => 100100 [9,3,2,1] => 300300 [9,3,1,1,1] => 600600 [9,2,2,2] => 450450 [9,2,2,1,1] => 900900 [9,2,1,1,1,1] => 1801800 [9,1,1,1,1,1,1] => 3603600 [8,7] => 6435 [8,6,1] => 45045 [8,5,2] => 135135 [8,5,1,1] => 270270 [8,4,3] => 225225 [8,4,2,1] => 675675 [8,4,1,1,1] => 1351350 [8,3,3,1] => 900900 [8,3,2,2] => 1351350 [8,3,2,1,1] => 2702700 [8,3,1,1,1,1] => 5405400 [8,2,2,2,1] => 4054050 [8,2,2,1,1,1] => 8108100 [7,7,1] => 51480 [7,6,2] => 180180 [7,6,1,1] => 360360 [7,5,3] => 360360 [7,5,2,1] => 1081080 [7,5,1,1,1] => 2162160 [7,4,4] => 450450 [7,4,3,1] => 1801800 [7,4,2,2] => 2702700 [7,4,2,1,1] => 5405400 [7,3,3,2] => 3603600 [7,3,3,1,1] => 7207200 [6,6,3] => 420420 [6,6,2,1] => 1261260 [6,6,1,1,1] => 2522520 [6,5,4] => 630630 [6,5,3,1] => 2522520 [6,5,2,2] => 3783780 [6,5,2,1,1] => 7567560 [6,4,4,1] => 3153150 [6,4,3,2] => 6306300 [6,3,3,3] => 8408400 [5,5,5] => 756756 [5,5,4,1] => 3783780 [5,5,3,2] => 7567560 [5,4,4,2] => 9459450 ----------------------------------------------------------------------------- Created: Mar 25, 2013 at 10:25 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Oct 29, 2017 at 20:26 by Martin Rubey