***************************************************************************** * 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: St000517 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The Kreweras number of an integer partition. This is defined for $\lambda \vdash n$ with $k$ parts as $$\frac{1}{n+1}\binom{n+1}{n+1-k,\mu_1(\lambda),\ldots,\mu_n(\lambda)}$$ where $\mu_j(\lambda)$ denotes the number of parts of $\lambda$ equal to $j$, see [1]. This formula indeed counts the number of noncrossing set partitions where the ordered block sizes are the partition $\lambda$. These numbers refine the Narayana numbers $N(n,k) = \frac{1}{k}\binom{n-1}{k-1}\binom{n}{k-1}$ and thus sum up to the Catalan numbers $\frac{1}{n+1}\binom{2n}{n}$. ----------------------------------------------------------------------------- References: [1] Reiner, V., Sommers, E. Weyl group $q$-Kreweras numbers and cyclic sieving [[arXiv:1605.09172]] ----------------------------------------------------------------------------- Code: def statistic(la): la = list(la) n = sum(la) k = len(la) multi = [n+1-k]+[ la.count(j) for j in [1..n] ] return multinomial(multi)/(n+1) ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 1 [2] => 1 [1,1] => 1 [3] => 1 [2,1] => 3 [1,1,1] => 1 [4] => 1 [3,1] => 4 [2,2] => 2 [2,1,1] => 6 [1,1,1,1] => 1 [5] => 1 [4,1] => 5 [3,2] => 5 [3,1,1] => 10 [2,2,1] => 10 [2,1,1,1] => 10 [1,1,1,1,1] => 1 [6] => 1 [5,1] => 6 [4,2] => 6 [4,1,1] => 15 [3,3] => 3 [3,2,1] => 30 [3,1,1,1] => 20 [2,2,2] => 5 [2,2,1,1] => 30 [2,1,1,1,1] => 15 [1,1,1,1,1,1] => 1 [7] => 1 [6,1] => 7 [5,2] => 7 [5,1,1] => 21 [4,3] => 7 [4,2,1] => 42 [4,1,1,1] => 35 [3,3,1] => 21 [3,2,2] => 21 [3,2,1,1] => 105 [3,1,1,1,1] => 35 [2,2,2,1] => 35 [2,2,1,1,1] => 70 [2,1,1,1,1,1] => 21 [1,1,1,1,1,1,1] => 1 [8] => 1 [7,1] => 8 [6,2] => 8 [6,1,1] => 28 [5,3] => 8 [5,2,1] => 56 [5,1,1,1] => 56 [4,4] => 4 [4,3,1] => 56 [4,2,2] => 28 [4,2,1,1] => 168 [4,1,1,1,1] => 70 [3,3,2] => 28 [3,3,1,1] => 84 [3,2,2,1] => 168 [3,2,1,1,1] => 280 [3,1,1,1,1,1] => 56 [2,2,2,2] => 14 [2,2,2,1,1] => 140 [2,2,1,1,1,1] => 140 [2,1,1,1,1,1,1] => 28 [1,1,1,1,1,1,1,1] => 1 [9] => 1 [8,1] => 9 [7,2] => 9 [7,1,1] => 36 [6,3] => 9 [6,2,1] => 72 [6,1,1,1] => 84 [5,4] => 9 [5,3,1] => 72 [5,2,2] => 36 [5,2,1,1] => 252 [5,1,1,1,1] => 126 [4,4,1] => 36 [4,3,2] => 72 [4,3,1,1] => 252 [4,2,2,1] => 252 [4,2,1,1,1] => 504 [4,1,1,1,1,1] => 126 [3,3,3] => 12 [3,3,2,1] => 252 [3,3,1,1,1] => 252 [3,2,2,2] => 84 [3,2,2,1,1] => 756 [3,2,1,1,1,1] => 630 [3,1,1,1,1,1,1] => 84 [2,2,2,2,1] => 126 [2,2,2,1,1,1] => 420 [2,2,1,1,1,1,1] => 252 [2,1,1,1,1,1,1,1] => 36 [1,1,1,1,1,1,1,1,1] => 1 [10] => 1 [9,1] => 10 [8,2] => 10 [8,1,1] => 45 [7,3] => 10 [7,2,1] => 90 [7,1,1,1] => 120 [6,4] => 10 [6,3,1] => 90 [6,2,2] => 45 [6,2,1,1] => 360 [6,1,1,1,1] => 210 [5,5] => 5 [5,4,1] => 90 [5,3,2] => 90 [5,3,1,1] => 360 [5,2,2,1] => 360 [5,2,1,1,1] => 840 [5,1,1,1,1,1] => 252 [4,4,2] => 45 [4,4,1,1] => 180 [4,3,3] => 45 [4,3,2,1] => 720 [4,3,1,1,1] => 840 [4,2,2,2] => 120 [4,2,2,1,1] => 1260 [4,2,1,1,1,1] => 1260 [4,1,1,1,1,1,1] => 210 [3,3,3,1] => 120 [3,3,2,2] => 180 [3,3,2,1,1] => 1260 [3,3,1,1,1,1] => 630 [3,2,2,2,1] => 840 [3,2,2,1,1,1] => 2520 [3,2,1,1,1,1,1] => 1260 [3,1,1,1,1,1,1,1] => 120 [2,2,2,2,2] => 42 [2,2,2,2,1,1] => 630 [2,2,2,1,1,1,1] => 1050 [2,2,1,1,1,1,1,1] => 420 [2,1,1,1,1,1,1,1,1] => 45 [1,1,1,1,1,1,1,1,1,1] => 1 [11] => 1 [10,1] => 11 [9,2] => 11 [9,1,1] => 55 [8,3] => 11 [8,2,1] => 110 [8,1,1,1] => 165 [7,4] => 11 [7,3,1] => 110 [7,2,2] => 55 [7,2,1,1] => 495 [7,1,1,1,1] => 330 [6,5] => 11 [6,4,1] => 110 [6,3,2] => 110 [6,3,1,1] => 495 [6,2,2,1] => 495 [6,2,1,1,1] => 1320 [6,1,1,1,1,1] => 462 [5,5,1] => 55 [5,4,2] => 110 [5,4,1,1] => 495 [5,3,3] => 55 [5,3,2,1] => 990 [5,3,1,1,1] => 1320 [5,2,2,2] => 165 [5,2,2,1,1] => 1980 [5,2,1,1,1,1] => 2310 [5,1,1,1,1,1,1] => 462 [4,4,3] => 55 [4,4,2,1] => 495 [4,4,1,1,1] => 660 [4,3,3,1] => 495 [4,3,2,2] => 495 [4,3,2,1,1] => 3960 [4,3,1,1,1,1] => 2310 [4,2,2,2,1] => 1320 [4,2,2,1,1,1] => 4620 [4,2,1,1,1,1,1] => 2772 [4,1,1,1,1,1,1,1] => 330 [3,3,3,2] => 165 [3,3,3,1,1] => 660 [3,3,2,2,1] => 1980 [3,3,2,1,1,1] => 4620 [3,3,1,1,1,1,1] => 1386 [3,2,2,2,2] => 330 [3,2,2,2,1,1] => 4620 [3,2,2,1,1,1,1] => 6930 [3,2,1,1,1,1,1,1] => 2310 [3,1,1,1,1,1,1,1,1] => 165 [2,2,2,2,2,1] => 462 [2,2,2,2,1,1,1] => 2310 [2,2,2,1,1,1,1,1] => 2310 [2,2,1,1,1,1,1,1,1] => 660 [2,1,1,1,1,1,1,1,1,1] => 55 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 1 [11,1] => 12 [10,2] => 12 [10,1,1] => 66 [9,3] => 12 [9,2,1] => 132 [9,1,1,1] => 220 [8,4] => 12 [8,3,1] => 132 [8,2,2] => 66 [8,2,1,1] => 660 [8,1,1,1,1] => 495 [7,5] => 12 [7,4,1] => 132 [7,3,2] => 132 [7,3,1,1] => 660 [7,2,2,1] => 660 [7,2,1,1,1] => 1980 [7,1,1,1,1,1] => 792 [6,6] => 6 [6,5,1] => 132 [6,4,2] => 132 [6,4,1,1] => 660 [6,3,3] => 66 [6,3,2,1] => 1320 [6,3,1,1,1] => 1980 [6,2,2,2] => 220 [6,2,2,1,1] => 2970 [6,2,1,1,1,1] => 3960 [6,1,1,1,1,1,1] => 924 [5,5,2] => 66 [5,5,1,1] => 330 [5,4,3] => 132 [5,4,2,1] => 1320 [5,4,1,1,1] => 1980 [5,3,3,1] => 660 [5,3,2,2] => 660 [5,3,2,1,1] => 5940 [5,3,1,1,1,1] => 3960 [5,2,2,2,1] => 1980 [5,2,2,1,1,1] => 7920 [5,2,1,1,1,1,1] => 5544 [5,1,1,1,1,1,1,1] => 792 [4,4,4] => 22 [4,4,3,1] => 660 [4,4,2,2] => 330 [4,4,2,1,1] => 2970 [4,4,1,1,1,1] => 1980 [4,3,3,2] => 660 [4,3,3,1,1] => 2970 [4,3,2,2,1] => 5940 [4,3,2,1,1,1] => 15840 [4,3,1,1,1,1,1] => 5544 [4,2,2,2,2] => 495 [4,2,2,2,1,1] => 7920 [4,2,2,1,1,1,1] => 13860 [4,2,1,1,1,1,1,1] => 5544 [4,1,1,1,1,1,1,1,1] => 495 [3,3,3,3] => 55 [3,3,3,2,1] => 1980 [3,3,3,1,1,1] => 2640 [3,3,2,2,2] => 990 [3,3,2,2,1,1] => 11880 [3,3,2,1,1,1,1] => 13860 [3,3,1,1,1,1,1,1] => 2772 [3,2,2,2,2,1] => 3960 [3,2,2,2,1,1,1] => 18480 [3,2,2,1,1,1,1,1] => 16632 [3,2,1,1,1,1,1,1,1] => 3960 [3,1,1,1,1,1,1,1,1,1] => 220 [2,2,2,2,2,2] => 132 [2,2,2,2,2,1,1] => 2772 [2,2,2,2,1,1,1,1] => 6930 [2,2,2,1,1,1,1,1,1] => 4620 [2,2,1,1,1,1,1,1,1,1] => 990 [2,1,1,1,1,1,1,1,1,1,1] => 66 [1,1,1,1,1,1,1,1,1,1,1,1] => 1 [13] => 1 [12,1] => 13 [10,3] => 13 [8,5] => 13 [7,6] => 13 [7,5,1] => 156 [7,4,2] => 156 [6,6,1] => 78 [6,4,2,1] => 1716 [5,5,3] => 78 [5,4,4] => 78 [5,4,3,1] => 1716 [5,4,2,2] => 858 [5,4,2,1,1] => 8580 [5,4,1,1,1,1] => 6435 [5,3,3,2] => 858 [5,3,3,1,1] => 4290 [5,3,2,2,1] => 8580 [5,3,2,1,1,1] => 25740 [4,4,4,1] => 286 [4,4,3,2] => 858 [4,4,3,1,1] => 4290 [4,4,2,2,1] => 4290 [4,3,3,3] => 286 [4,3,3,2,1] => 8580 [3,3,3,3,1] => 715 [3,3,3,2,2] => 1430 [3,3,2,2,2,1] => 12870 [3,2,2,2,2,2] => 1287 [2,2,2,2,2,2,1] => 1716 [1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [14] => 1 [13,1] => 14 [9,5] => 14 [8,5,1] => 182 [7,7] => 7 [7,5,2] => 182 [7,4,3] => 182 [6,6,2] => 91 [6,4,4] => 91 [6,2,2,2,2] => 1001 [5,5,4] => 91 [5,5,1,1,1,1] => 5005 [5,4,3,2] => 2184 [5,4,3,1,1] => 12012 [5,4,2,2,1] => 12012 [5,4,2,1,1,1] => 40040 [5,3,3,3] => 364 [5,3,3,2,1] => 12012 [5,2,2,2,2,1] => 10010 [4,4,4,2] => 364 [4,4,3,3] => 546 [4,4,3,2,1] => 12012 [4,3,2,2,2,1] => 40040 [3,3,3,3,2] => 1001 [3,3,3,3,1,1] => 5005 [3,3,2,2,2,2] => 5005 [2,2,2,2,2,2,2] => 429 [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [15] => 1 [14,1] => 15 [9,5,1] => 210 [8,5,2] => 210 [7,5,3] => 210 [6,6,3] => 105 [6,5,4] => 210 [6,5,1,1,1,1] => 15015 [6,3,3,3] => 455 [6,2,2,2,2,1] => 15015 [5,5,5] => 35 [5,4,3,3] => 1365 [5,4,3,2,1] => 32760 [5,4,3,1,1,1] => 60060 [5,3,2,2,2,1] => 60060 [4,4,4,3] => 455 [4,4,4,1,1,1] => 10010 [4,3,3,3,2] => 5460 [3,3,3,3,3] => 273 [3,3,3,3,2,1] => 15015 [3,3,3,2,2,2] => 10010 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [16] => 1 [15,1] => 16 [8,8] => 8 [8,5,3] => 240 [7,5,3,1] => 3360 [6,6,4] => 120 [5,5,3,3] => 840 [5,5,2,2,2] => 3640 [5,4,4,3] => 1680 [5,4,3,2,1,1] => 262080 [5,4,2,2,2,1] => 87360 [4,4,4,4] => 140 [4,4,4,2,2] => 3640 [4,4,3,3,2] => 10920 [4,3,3,3,3] => 1820 [4,3,3,3,2,1] => 87360 [3,3,3,3,2,2] => 10920 [2,2,2,2,2,2,2,2] => 1430 [17] => 1 [8,6,3] => 272 [7,5,3,2] => 4080 [6,5,3,3] => 2040 [6,5,2,2,2] => 9520 [6,4,4,3] => 2040 [6,4,4,1,1,1] => 61880 [6,3,3,3,2] => 9520 [6,3,3,3,1,1] => 61880 [5,5,4,3] => 2040 [5,5,4,1,1,1] => 61880 [5,5,2,2,2,1] => 61880 [5,4,4,4] => 680 [5,4,3,2,2,1] => 371280 [5,3,3,3,2,1] => 123760 [4,4,4,3,2] => 9520 [4,4,4,3,1,1] => 61880 [4,4,4,2,2,1] => 61880 [4,4,3,3,3] => 4760 [3,3,3,3,3,2] => 6188 [4,4,4,3,2,1] => 171360 [5,4,3,3,2,1] => 514080 [6,3,3,3,2,1] => 171360 [6,5,2,2,2,1] => 171360 [5,5,3,3,1,1] => 128520 [6,5,4,1,1,1] => 171360 [5,5,3,3,2] => 18360 [5,5,4,2,2] => 18360 [6,4,4,2,2] => 18360 [6,5,4,3] => 4896 [4,4,4,3,3] => 6120 [4,4,4,4,2] => 3060 [5,5,4,4] => 1224 [2,2,2,2,2,2,2,2,2] => 4862 [3,3,3,3,3,3] => 1428 [18] => 1 [6,6,6] => 51 [9,6,3] => 306 [8,6,4] => 306 [9,9] => 9 [5,4,4,3,2,1] => 697680 [5,5,3,3,2,1] => 348840 [5,5,4,2,2,1] => 348840 [6,4,4,2,2,1] => 348840 [5,5,4,3,1,1] => 348840 [6,4,4,3,1,1] => 348840 [6,5,3,3,1,1] => 348840 [5,5,4,3,2] => 46512 [6,4,4,3,2] => 46512 [6,5,3,3,2] => 46512 [6,5,4,2,2] => 46512 [6,5,4,3,1] => 93024 [6,5,4,1,1,1,1] => 813960 [4,4,4,4,3] => 3876 [4,3,3,3,3,3] => 11628 [19] => 1 [9,6,4] => 342 [8,5,4,2] => 5814 [8,5,5,1] => 2907 [5,5,4,3,2,1] => 930240 [6,4,4,3,2,1] => 930240 [6,5,3,3,2,1] => 930240 [6,5,4,2,2,1] => 930240 [6,5,4,3,1,1] => 930240 [6,5,4,3,2] => 116280 [6,5,2,2,2,2,1] => 1162800 [6,5,4,2,1,1,1] => 4651200 [7,5,4,3,1] => 116280 [3,3,3,3,3,3,2] => 38760 [4,4,3,3,3,3] => 38760 [4,4,4,4,4] => 969 [5,5,5,5] => 285 [20] => 1 [8,6,4,2] => 6840 [10,6,4] => 380 [10,7,3] => 380 [9,7,4] => 380 [9,5,5,1] => 3420 [6,5,4,3,2,1] => 2441880 [6,3,3,3,3,2,1] => 1627920 [6,5,3,2,2,2,1] => 6511680 [6,5,4,3,1,1,1] => 6511680 [3,3,3,3,3,3,3] => 7752 [4,4,4,3,3,3] => 67830 [21] => 1 [11,7,3] => 420 [4,4,4,4,3,2,1] => 2238390 [6,4,3,3,3,2,1] => 8953560 [6,5,4,2,2,2,1] => 8953560 [6,5,4,3,2,1,1] => 26860680 [4,4,4,4,3,3] => 65835 [9,6,4,3] => 9240 [5,4,4,4,3,2,1] => 12113640 [6,5,3,3,3,2,1] => 12113640 [6,5,4,3,2,2,1] => 36340920 [9,6,5,3] => 10626 [8,6,5,3,1] => 212520 [6,4,4,4,3,2,1] => 16151520 [6,5,4,3,3,2,1] => 48454560 [3,3,3,3,3,3,3,3] => 43263 [4,4,4,4,4,4] => 7084 [11,7,5,1] => 12144 [9,7,5,3] => 12144 [8,8,8] => 92 [5,5,5,4,3,2,1] => 21252000 [6,5,4,4,3,2,1] => 63756000 [9,7,5,3,1] => 303600 [10,7,5,3] => 13800 [6,5,5,4,3,2,1] => 82882800 [9,7,5,4,1] => 358800 [6,6,5,4,3,2,1] => 106563600 [7,6,5,4,3,2] => 9687600 [3,3,3,3,3,3,3,3,3] => 246675 [7,6,5,4,3,2,1] => 271252800 [7,6,5,4,3,1,1,1] => 994593600 [10,7,6,4,1] => 491400 [9,7,6,4,2] => 491400 [10,8,5,4,1] => 491400 [7,6,5,4,2,2,2,1] => 1311055200 [10,8,6,4,1] => 570024 [9,7,5,5,3,1] => 8550360 [7,6,5,3,3,3,2,1] => 1710072000 [11,8,6,4,1] => 657720 [10,8,6,4,2] => 657720 [11,8,6,5,1] => 755160 [4,4,4,4,4,4,4,4] => 420732 [12,9,7,5,1] => 1113024 [13,9,7,5,1] => 1256640 [11,9,7,5,3,1] => 45239040 [11,8,7,5,4,1] => 45239040 [11,9,7,5,5,3] => 39480480 [11,9,7,7,5,3,3] => 1466110800 ----------------------------------------------------------------------------- Created: May 31, 2016 at 14:57 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Jun 19, 2023 at 10:40 by Martin Rubey