***************************************************************************** * 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 ----------------------------------------------------------------------------- Created: May 31, 2016 at 14:57 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Oct 29, 2017 at 21:33 by Martin Rubey