***************************************************************************** * 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: St000063 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of linear extensions of a certain poset defined for an integer partition. The poset is constructed in David Speyer's answer to Matt Fayers' question [3]. The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment. This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras. ----------------------------------------------------------------------------- References: [1] Fayers, M. Dyck tilings and the homogeneous Garnir relations for graded Specht modules [[arXiv:1309.6467]] [2] Kenyon, R. W., Wilson, D. B. Double-dimer pairings and skew Young diagrams [[MathSciNet:2811099]] [3] Fayers, M. A function from partitions to natural numbers - is it familiar? [[MathOverflow:132338]] ----------------------------------------------------------------------------- Code: def statistic( P ): if P.is_empty(): return 1 cells = P.cells() m = max( i+j for i,j in cells ) found_max = False while found_max is False: i,j = cells.pop() if i+j == m: found_max = True P1 = Partition( P[i+1:] ) P2 = Partition( P.conjugate()[j+1:] ).conjugate() return binomial(i+j+2,i+1)*statistic(P1)*statistic(P2) ----------------------------------------------------------------------------- Statistic values: [] => 1 [1] => 2 [2] => 3 [1,1] => 3 [3] => 4 [2,1] => 6 [1,1,1] => 4 [4] => 5 [3,1] => 8 [2,2] => 6 [2,1,1] => 8 [1,1,1,1] => 5 [5] => 6 [4,1] => 10 [3,2] => 12 [3,1,1] => 12 [2,2,1] => 12 [2,1,1,1] => 10 [1,1,1,1,1] => 6 [6] => 7 [5,1] => 12 [4,2] => 15 [4,1,1] => 15 [3,3] => 10 [3,2,1] => 24 [3,1,1,1] => 15 [2,2,2] => 10 [2,2,1,1] => 15 [2,1,1,1,1] => 12 [1,1,1,1,1,1] => 7 [7] => 8 [6,1] => 14 [5,2] => 18 [5,1,1] => 18 [4,3] => 20 [4,2,1] => 30 [4,1,1,1] => 20 [3,3,1] => 20 [3,2,2] => 20 [3,2,1,1] => 30 [3,1,1,1,1] => 18 [2,2,2,1] => 20 [2,2,1,1,1] => 18 [2,1,1,1,1,1] => 14 [1,1,1,1,1,1,1] => 8 [8] => 9 [7,1] => 16 [6,2] => 21 [6,1,1] => 21 [5,3] => 24 [5,2,1] => 36 [5,1,1,1] => 24 [4,4] => 15 [4,3,1] => 40 [4,2,2] => 30 [4,2,1,1] => 40 [4,1,1,1,1] => 24 [3,3,2] => 30 [3,3,1,1] => 30 [3,2,2,1] => 40 [3,2,1,1,1] => 36 [3,1,1,1,1,1] => 21 [2,2,2,2] => 15 [2,2,2,1,1] => 24 [2,2,1,1,1,1] => 21 [2,1,1,1,1,1,1] => 16 [1,1,1,1,1,1,1,1] => 9 [9] => 10 [8,1] => 18 [7,2] => 24 [7,1,1] => 24 [6,3] => 28 [6,2,1] => 42 [6,1,1,1] => 28 [5,4] => 30 [5,3,1] => 48 [5,2,2] => 36 [5,2,1,1] => 48 [5,1,1,1,1] => 30 [4,4,1] => 30 [4,3,2] => 60 [4,3,1,1] => 60 [4,2,2,1] => 60 [4,2,1,1,1] => 48 [4,1,1,1,1,1] => 28 [3,3,3] => 20 [3,3,2,1] => 60 [3,3,1,1,1] => 36 [3,2,2,2] => 30 [3,2,2,1,1] => 48 [3,2,1,1,1,1] => 42 [3,1,1,1,1,1,1] => 24 [2,2,2,2,1] => 30 [2,2,2,1,1,1] => 28 [2,2,1,1,1,1,1] => 24 [2,1,1,1,1,1,1,1] => 18 [1,1,1,1,1,1,1,1,1] => 10 [10] => 11 [9,1] => 20 [8,2] => 27 [8,1,1] => 27 [7,3] => 32 [7,2,1] => 48 [7,1,1,1] => 32 [6,4] => 35 [6,3,1] => 56 [6,2,2] => 42 [6,2,1,1] => 56 [6,1,1,1,1] => 35 [5,5] => 21 [5,4,1] => 60 [5,3,2] => 72 [5,3,1,1] => 72 [5,2,2,1] => 72 [5,2,1,1,1] => 60 [5,1,1,1,1,1] => 35 [4,4,2] => 45 [4,4,1,1] => 45 [4,3,3] => 40 [4,3,2,1] => 120 [4,3,1,1,1] => 72 [4,2,2,2] => 45 [4,2,2,1,1] => 72 [4,2,1,1,1,1] => 56 [4,1,1,1,1,1,1] => 32 [3,3,3,1] => 40 [3,3,2,2] => 45 [3,3,2,1,1] => 72 [3,3,1,1,1,1] => 42 [3,2,2,2,1] => 60 [3,2,2,1,1,1] => 56 [3,2,1,1,1,1,1] => 48 [3,1,1,1,1,1,1,1] => 27 [2,2,2,2,2] => 21 [2,2,2,2,1,1] => 35 [2,2,2,1,1,1,1] => 32 [2,2,1,1,1,1,1,1] => 27 [2,1,1,1,1,1,1,1,1] => 20 [1,1,1,1,1,1,1,1,1,1] => 11 [11] => 12 [10,1] => 22 [9,2] => 30 [9,1,1] => 30 [8,3] => 36 [8,2,1] => 54 [8,1,1,1] => 36 [7,4] => 40 [7,3,1] => 64 [7,2,2] => 48 [7,2,1,1] => 64 [7,1,1,1,1] => 40 [6,5] => 42 [6,4,1] => 70 [6,3,2] => 84 [6,3,1,1] => 84 [6,2,2,1] => 84 [6,2,1,1,1] => 70 [6,1,1,1,1,1] => 42 [5,5,1] => 42 [5,4,2] => 90 [5,4,1,1] => 90 [5,3,3] => 60 [5,3,2,1] => 144 [5,3,1,1,1] => 90 [5,2,2,2] => 60 [5,2,2,1,1] => 90 [5,2,1,1,1,1] => 70 [5,1,1,1,1,1,1] => 40 [4,4,3] => 60 [4,4,2,1] => 90 [4,4,1,1,1] => 60 [4,3,3,1] => 80 [4,3,2,2] => 90 [4,3,2,1,1] => 144 [4,3,1,1,1,1] => 84 [4,2,2,2,1] => 90 [4,2,2,1,1,1] => 84 [4,2,1,1,1,1,1] => 64 [4,1,1,1,1,1,1,1] => 36 [3,3,3,2] => 60 [3,3,3,1,1] => 60 [3,3,2,2,1] => 90 [3,3,2,1,1,1] => 84 [3,3,1,1,1,1,1] => 48 [3,2,2,2,2] => 42 [3,2,2,2,1,1] => 70 [3,2,2,1,1,1,1] => 64 [3,2,1,1,1,1,1,1] => 54 [3,1,1,1,1,1,1,1,1] => 30 [2,2,2,2,2,1] => 42 [2,2,2,2,1,1,1] => 40 [2,2,2,1,1,1,1,1] => 36 [2,2,1,1,1,1,1,1,1] => 30 [2,1,1,1,1,1,1,1,1,1] => 22 [1,1,1,1,1,1,1,1,1,1,1] => 12 [12] => 13 [11,1] => 24 [10,2] => 33 [10,1,1] => 33 [9,3] => 40 [9,2,1] => 60 [9,1,1,1] => 40 [8,4] => 45 [8,3,1] => 72 [8,2,2] => 54 [8,2,1,1] => 72 [8,1,1,1,1] => 45 [7,5] => 48 [7,4,1] => 80 [7,3,2] => 96 [7,3,1,1] => 96 [7,2,2,1] => 96 [7,2,1,1,1] => 80 [7,1,1,1,1,1] => 48 [6,6] => 28 [6,5,1] => 84 [6,4,2] => 105 [6,4,1,1] => 105 [6,3,3] => 70 [6,3,2,1] => 168 [6,3,1,1,1] => 105 [6,2,2,2] => 70 [6,2,2,1,1] => 105 [6,2,1,1,1,1] => 84 [6,1,1,1,1,1,1] => 48 [5,5,2] => 63 [5,5,1,1] => 63 [5,4,3] => 120 [5,4,2,1] => 180 [5,4,1,1,1] => 120 [5,3,3,1] => 120 [5,3,2,2] => 120 [5,3,2,1,1] => 180 [5,3,1,1,1,1] => 105 [5,2,2,2,1] => 120 [5,2,2,1,1,1] => 105 [5,2,1,1,1,1,1] => 80 [5,1,1,1,1,1,1,1] => 45 [4,4,4] => 35 [4,4,3,1] => 120 [4,4,2,2] => 90 [4,4,2,1,1] => 120 [4,4,1,1,1,1] => 70 [4,3,3,2] => 120 [4,3,3,1,1] => 120 [4,3,2,2,1] => 180 [4,3,2,1,1,1] => 168 [4,3,1,1,1,1,1] => 96 [4,2,2,2,2] => 63 [4,2,2,2,1,1] => 105 [4,2,2,1,1,1,1] => 96 [4,2,1,1,1,1,1,1] => 72 [4,1,1,1,1,1,1,1,1] => 40 [3,3,3,3] => 35 [3,3,3,2,1] => 120 [3,3,3,1,1,1] => 70 [3,3,2,2,2] => 63 [3,3,2,2,1,1] => 105 [3,3,2,1,1,1,1] => 96 [3,3,1,1,1,1,1,1] => 54 [3,2,2,2,2,1] => 84 [3,2,2,2,1,1,1] => 80 [3,2,2,1,1,1,1,1] => 72 [3,2,1,1,1,1,1,1,1] => 60 [3,1,1,1,1,1,1,1,1,1] => 33 [2,2,2,2,2,2] => 28 [2,2,2,2,2,1,1] => 48 [2,2,2,2,1,1,1,1] => 45 [2,2,2,1,1,1,1,1,1] => 40 [2,2,1,1,1,1,1,1,1,1] => 33 [2,1,1,1,1,1,1,1,1,1,1] => 24 [1,1,1,1,1,1,1,1,1,1,1,1] => 13 [5,4,3,1] => 240 [5,4,2,2] => 180 [5,4,2,1,1] => 240 [5,3,3,2] => 180 [5,3,3,1,1] => 180 [5,3,2,2,1] => 240 [4,4,3,2] => 180 [4,4,3,1,1] => 180 [4,4,2,2,1] => 180 [4,3,3,2,1] => 240 [5,4,3,2] => 360 [5,4,3,1,1] => 360 [5,4,2,2,1] => 360 [5,3,3,2,1] => 360 [4,4,3,2,1] => 360 [5,4,3,2,1] => 720 ----------------------------------------------------------------------------- Created: May 31, 2013 at 11:49 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Mar 19, 2019 at 23:37 by Martin Rubey