***************************************************************************** * 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: St000618 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of self-evacuating tableaux of given shape. This is the same as the number of standard domino tableaux of the given shape. ----------------------------------------------------------------------------- References: [1] Stembridge, J. R. Canonical bases and self-evacuating tableaux [[MathSciNet:1387685]] ----------------------------------------------------------------------------- Code: def statistic_alt(la): chi = SymmetricGroupRepresentation(la) r = abs(ZZ(chi.to_character()(Permutation(range(la.size(),0,-1))))) assert r==statistic(la) return r def statistic(la): n = la.size() la = la + [0]*(n-len(la)) E_la = sorted([la[j] + n-j-1 for j in range(n) if is_even(la[j] + n-j-1)], reverse=True) rE = len(E_la) la_e = [ZZ(E_la[i]/2-rE+i+1) for i in range(rE)] O_la = sorted([la[j] + n-j-2 for j in range(n) if is_even(la[j] +n-j-2)], reverse=True) rO = len(O_la) la_o = [ZZ(O_la[i]/2-rO+i+1) for i in range(rO)] if abs(rE - rO)<=1: return (binomial(n//2, sum(la_e))* StandardTableaux(la_e).cardinality()* StandardTableaux(la_o).cardinality()) else: return 0 ----------------------------------------------------------------------------- Statistic values: [1] => 1 [2] => 1 [1,1] => 1 [3] => 1 [2,1] => 0 [1,1,1] => 1 [4] => 1 [3,1] => 1 [2,2] => 2 [2,1,1] => 1 [1,1,1,1] => 1 [5] => 1 [4,1] => 0 [3,2] => 1 [3,1,1] => 2 [2,2,1] => 1 [2,1,1,1] => 0 [1,1,1,1,1] => 1 [6] => 1 [5,1] => 1 [4,2] => 3 [4,1,1] => 2 [3,3] => 3 [3,2,1] => 0 [3,1,1,1] => 2 [2,2,2] => 3 [2,2,1,1] => 3 [2,1,1,1,1] => 1 [1,1,1,1,1,1] => 1 [7] => 1 [6,1] => 0 [5,2] => 2 [5,1,1] => 3 [4,3] => 0 [4,2,1] => 1 [4,1,1,1] => 0 [3,3,1] => 3 [3,2,2] => 3 [3,2,1,1] => 1 [3,1,1,1,1] => 3 [2,2,2,1] => 0 [2,2,1,1,1] => 2 [2,1,1,1,1,1] => 0 [1,1,1,1,1,1,1] => 1 [8] => 1 [7,1] => 1 [6,2] => 4 [6,1,1] => 3 [5,3] => 4 [5,2,1] => 0 [5,1,1,1] => 3 [4,4] => 6 [4,3,1] => 2 [4,2,2] => 8 [4,2,1,1] => 6 [4,1,1,1,1] => 3 [3,3,2] => 6 [3,3,1,1] => 8 [3,2,2,1] => 2 [3,2,1,1,1] => 0 [3,1,1,1,1,1] => 3 [2,2,2,2] => 6 [2,2,2,1,1] => 4 [2,2,1,1,1,1] => 4 [2,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1] => 1 [9] => 1 [8,1] => 0 [7,2] => 3 [7,1,1] => 4 [6,3] => 0 [6,2,1] => 1 [6,1,1,1] => 0 [5,4] => 2 [5,3,1] => 6 [5,2,2] => 8 [5,2,1,1] => 3 [5,1,1,1,1] => 6 [4,4,1] => 4 [4,3,2] => 0 [4,3,1,1] => 0 [4,2,2,1] => 0 [4,2,1,1,1] => 3 [4,1,1,1,1,1] => 0 [3,3,3] => 6 [3,3,2,1] => 0 [3,3,1,1,1] => 8 [3,2,2,2] => 4 [3,2,2,1,1] => 6 [3,2,1,1,1,1] => 1 [3,1,1,1,1,1,1] => 4 [2,2,2,2,1] => 2 [2,2,2,1,1,1] => 0 [2,2,1,1,1,1,1] => 3 [2,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1] => 1 [10] => 1 [9,1] => 1 [8,2] => 5 [8,1,1] => 4 [7,3] => 5 [7,2,1] => 0 [7,1,1,1] => 4 [6,4] => 10 [6,3,1] => 5 [6,2,2] => 15 [6,2,1,1] => 10 [6,1,1,1,1] => 6 [5,5] => 10 [5,4,1] => 0 [5,3,2] => 10 [5,3,1,1] => 15 [5,2,2,1] => 5 [5,2,1,1,1] => 0 [5,1,1,1,1,1] => 6 [4,4,2] => 20 [4,4,1,1] => 20 [4,3,3] => 10 [4,3,2,1] => 0 [4,3,1,1,1] => 5 [4,2,2,2] => 20 [4,2,2,1,1] => 15 [4,2,1,1,1,1] => 10 [4,1,1,1,1,1,1] => 4 [3,3,3,1] => 10 [3,3,2,2] => 20 [3,3,2,1,1] => 10 [3,3,1,1,1,1] => 15 [3,2,2,2,1] => 0 [3,2,2,1,1,1] => 5 [3,2,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1] => 4 [2,2,2,2,2] => 10 [2,2,2,2,1,1] => 10 [2,2,2,1,1,1,1] => 5 [2,2,1,1,1,1,1,1] => 5 [2,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1,1] => 1 [11] => 1 [10,1] => 0 [9,2] => 4 [9,1,1] => 5 [8,3] => 0 [8,2,1] => 1 [8,1,1,1] => 0 [7,4] => 5 [7,3,1] => 10 [7,2,2] => 15 [7,2,1,1] => 6 [7,1,1,1,1] => 10 [6,5] => 0 [6,4,1] => 5 [6,3,2] => 0 [6,3,1,1] => 0 [6,2,2,1] => 0 [6,2,1,1,1] => 4 [6,1,1,1,1,1] => 0 [5,5,1] => 10 [5,4,2] => 10 [5,4,1,1] => 5 [5,3,3] => 20 [5,3,2,1] => 0 [5,3,1,1,1] => 20 [5,2,2,2] => 15 [5,2,2,1,1] => 20 [5,2,1,1,1,1] => 4 [5,1,1,1,1,1,1] => 10 [4,4,3] => 10 [4,4,2,1] => 0 [4,4,1,1,1] => 15 [4,3,3,1] => 0 [4,3,2,2] => 0 [4,3,2,1,1] => 0 [4,3,1,1,1,1] => 0 [4,2,2,2,1] => 5 [4,2,2,1,1,1] => 0 [4,2,1,1,1,1,1] => 6 [4,1,1,1,1,1,1,1] => 0 [3,3,3,2] => 10 [3,3,3,1,1] => 20 [3,3,2,2,1] => 10 [3,3,2,1,1,1] => 0 [3,3,1,1,1,1,1] => 15 [3,2,2,2,2] => 10 [3,2,2,2,1,1] => 5 [3,2,2,1,1,1,1] => 10 [3,2,1,1,1,1,1,1] => 1 [3,1,1,1,1,1,1,1,1] => 5 [2,2,2,2,2,1] => 0 [2,2,2,2,1,1,1] => 5 [2,2,2,1,1,1,1,1] => 0 [2,2,1,1,1,1,1,1,1] => 4 [2,1,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 1 [11,1] => 1 [10,2] => 6 [10,1,1] => 5 [9,3] => 6 [9,2,1] => 0 [9,1,1,1] => 5 [8,4] => 15 [8,3,1] => 9 [8,2,2] => 24 [8,2,1,1] => 15 [8,1,1,1,1] => 10 [7,5] => 15 [7,4,1] => 0 [7,3,2] => 15 [7,3,1,1] => 24 [7,2,2,1] => 9 [7,2,1,1,1] => 0 [7,1,1,1,1,1] => 10 [6,6] => 20 [6,5,1] => 5 [6,4,2] => 45 [6,4,1,1] => 40 [6,3,3] => 30 [6,3,2,1] => 0 [6,3,1,1,1] => 16 [6,2,2,2] => 45 [6,2,2,1,1] => 36 [6,2,1,1,1,1] => 20 [6,1,1,1,1,1,1] => 10 [5,5,2] => 40 [5,5,1,1] => 45 [5,4,3] => 0 [5,4,2,1] => 5 [5,4,1,1,1] => 0 [5,3,3,1] => 30 [5,3,2,2] => 45 [5,3,2,1,1] => 20 [5,3,1,1,1,1] => 36 [5,2,2,2,1] => 0 [5,2,2,1,1,1] => 16 [5,2,1,1,1,1,1] => 0 [5,1,1,1,1,1,1,1] => 10 [4,4,4] => 30 [4,4,3,1] => 30 [4,4,2,2] => 80 [4,4,2,1,1] => 45 [4,4,1,1,1,1] => 45 [4,3,3,2] => 30 [4,3,3,1,1] => 30 [4,3,2,2,1] => 5 [4,3,2,1,1,1] => 0 [4,3,1,1,1,1,1] => 9 [4,2,2,2,2] => 45 [4,2,2,2,1,1] => 40 [4,2,2,1,1,1,1] => 24 [4,2,1,1,1,1,1,1] => 15 [4,1,1,1,1,1,1,1,1] => 5 [3,3,3,3] => 30 [3,3,3,2,1] => 0 [3,3,3,1,1,1] => 30 [3,3,2,2,2] => 40 [3,3,2,2,1,1] => 45 [3,3,2,1,1,1,1] => 15 [3,3,1,1,1,1,1,1] => 24 [3,2,2,2,2,1] => 5 [3,2,2,2,1,1,1] => 0 [3,2,2,1,1,1,1,1] => 9 [3,2,1,1,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1,1,1] => 5 [2,2,2,2,2,2] => 20 [2,2,2,2,2,1,1] => 15 [2,2,2,2,1,1,1,1] => 15 [2,2,2,1,1,1,1,1,1] => 6 [2,2,1,1,1,1,1,1,1,1] => 6 [2,1,1,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1,1,1,1] => 1 ----------------------------------------------------------------------------- Created: Sep 26, 2016 at 23:00 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Sep 26, 2016 at 23:00 by Martin Rubey