***************************************************************************** * 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: St000477 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The weight of a partition according to Alladi. ----------------------------------------------------------------------------- References: [1] Alladi, K. Partition identities involving gaps and weights [[MathSciNet:1401759]] ----------------------------------------------------------------------------- Code: def statistic(pi): """ The weight according to Alladi. sage: r=8; RR = [pi for pi in Partitions(r) if all(pi[i] - pi[i+1] >= 2 for i in range(len(pi)-1))] sage: sum(weight(pi) for pi in RR) == Partitions(r).cardinality() """ return pi[-1]*prod(pi[i] - pi[i+1] -1 for i in range(len(pi)-1)) ----------------------------------------------------------------------------- Statistic values: [2] => 2 [1,1] => -1 [3] => 3 [2,1] => 0 [1,1,1] => 1 [4] => 4 [3,1] => 1 [2,2] => -2 [2,1,1] => 0 [1,1,1,1] => -1 [5] => 5 [4,1] => 2 [3,2] => 0 [3,1,1] => -1 [2,2,1] => 0 [2,1,1,1] => 0 [1,1,1,1,1] => 1 [6] => 6 [5,1] => 3 [4,2] => 2 [4,1,1] => -2 [3,3] => -3 [3,2,1] => 0 [3,1,1,1] => 1 [2,2,2] => 2 [2,2,1,1] => 0 [2,1,1,1,1] => 0 [1,1,1,1,1,1] => -1 [7] => 7 [6,1] => 4 [5,2] => 4 [5,1,1] => -3 [4,3] => 0 [4,2,1] => 0 [4,1,1,1] => 2 [3,3,1] => -1 [3,2,2] => 0 [3,2,1,1] => 0 [3,1,1,1,1] => -1 [2,2,2,1] => 0 [2,2,1,1,1] => 0 [2,1,1,1,1,1] => 0 [1,1,1,1,1,1,1] => 1 [8] => 8 [7,1] => 5 [6,2] => 6 [6,1,1] => -4 [5,3] => 3 [5,2,1] => 0 [5,1,1,1] => 3 [4,4] => -4 [4,3,1] => 0 [4,2,2] => -2 [4,2,1,1] => 0 [4,1,1,1,1] => -2 [3,3,2] => 0 [3,3,1,1] => 1 [3,2,2,1] => 0 [3,2,1,1,1] => 0 [3,1,1,1,1,1] => 1 [2,2,2,2] => -2 [2,2,2,1,1] => 0 [2,2,1,1,1,1] => 0 [2,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1] => -1 [9] => 9 [8,1] => 6 [7,2] => 8 [7,1,1] => -5 [6,3] => 6 [6,2,1] => 0 [6,1,1,1] => 4 [5,4] => 0 [5,3,1] => 1 [5,2,2] => -4 [5,2,1,1] => 0 [5,1,1,1,1] => -3 [4,4,1] => -2 [4,3,2] => 0 [4,3,1,1] => 0 [4,2,2,1] => 0 [4,2,1,1,1] => 0 [4,1,1,1,1,1] => 2 [3,3,3] => 3 [3,3,2,1] => 0 [3,3,1,1,1] => -1 [3,2,2,2] => 0 [3,2,2,1,1] => 0 [3,2,1,1,1,1] => 0 [3,1,1,1,1,1,1] => -1 [2,2,2,2,1] => 0 [2,2,2,1,1,1] => 0 [2,2,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1] => 1 [10] => 10 [9,1] => 7 [8,2] => 10 [8,1,1] => -6 [7,3] => 9 [7,2,1] => 0 [7,1,1,1] => 5 [6,4] => 4 [6,3,1] => 2 [6,2,2] => -6 [6,2,1,1] => 0 [6,1,1,1,1] => -4 [5,5] => -5 [5,4,1] => 0 [5,3,2] => 0 [5,3,1,1] => -1 [5,2,2,1] => 0 [5,2,1,1,1] => 0 [5,1,1,1,1,1] => 3 [4,4,2] => -2 [4,4,1,1] => 2 [4,3,3] => 0 [4,3,2,1] => 0 [4,3,1,1,1] => 0 [4,2,2,2] => 2 [4,2,2,1,1] => 0 [4,2,1,1,1,1] => 0 [4,1,1,1,1,1,1] => -2 [3,3,3,1] => 1 [3,3,2,2] => 0 [3,3,2,1,1] => 0 [3,3,1,1,1,1] => 1 [3,2,2,2,1] => 0 [3,2,2,1,1,1] => 0 [3,2,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1] => 1 [2,2,2,2,2] => 2 [2,2,2,2,1,1] => 0 [2,2,2,1,1,1,1] => 0 [2,2,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1] => -1 [11] => 11 [10,1] => 8 [9,2] => 12 [9,1,1] => -7 [8,3] => 12 [8,2,1] => 0 [8,1,1,1] => 6 [7,4] => 8 [7,3,1] => 3 [7,2,2] => -8 [7,2,1,1] => 0 [7,1,1,1,1] => -5 [6,5] => 0 [6,4,1] => 2 [6,3,2] => 0 [6,3,1,1] => -2 [6,2,2,1] => 0 [6,2,1,1,1] => 0 [6,1,1,1,1,1] => 4 [5,5,1] => -3 [5,4,2] => 0 [5,4,1,1] => 0 [5,3,3] => -3 [5,3,2,1] => 0 [5,3,1,1,1] => 1 [5,2,2,2] => 4 [5,2,2,1,1] => 0 [5,2,1,1,1,1] => 0 [5,1,1,1,1,1,1] => -3 [4,4,3] => 0 [4,4,2,1] => 0 [4,4,1,1,1] => -2 [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] => 0 [4,2,2,1,1,1] => 0 [4,2,1,1,1,1,1] => 0 [4,1,1,1,1,1,1,1] => 2 [3,3,3,2] => 0 [3,3,3,1,1] => -1 [3,3,2,2,1] => 0 [3,3,2,1,1,1] => 0 [3,3,1,1,1,1,1] => -1 [3,2,2,2,2] => 0 [3,2,2,2,1,1] => 0 [3,2,2,1,1,1,1] => 0 [3,2,1,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1,1] => -1 [2,2,2,2,2,1] => 0 [2,2,2,2,1,1,1] => 0 [2,2,2,1,1,1,1,1] => 0 [2,2,1,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 12 [11,1] => 9 [10,2] => 14 [10,1,1] => -8 [9,3] => 15 [9,2,1] => 0 [9,1,1,1] => 7 [8,4] => 12 [8,3,1] => 4 [8,2,2] => -10 [8,2,1,1] => 0 [8,1,1,1,1] => -6 [7,5] => 5 [7,4,1] => 4 [7,3,2] => 0 [7,3,1,1] => -3 [7,2,2,1] => 0 [7,2,1,1,1] => 0 [7,1,1,1,1,1] => 5 [6,6] => -6 [6,5,1] => 0 [6,4,2] => 2 [6,4,1,1] => -2 [6,3,3] => -6 [6,3,2,1] => 0 [6,3,1,1,1] => 2 [6,2,2,2] => 6 [6,2,2,1,1] => 0 [6,2,1,1,1,1] => 0 [6,1,1,1,1,1,1] => -4 [5,5,2] => -4 [5,5,1,1] => 3 [5,4,3] => 0 [5,4,2,1] => 0 [5,4,1,1,1] => 0 [5,3,3,1] => -1 [5,3,2,2] => 0 [5,3,2,1,1] => 0 [5,3,1,1,1,1] => -1 [5,2,2,2,1] => 0 [5,2,2,1,1,1] => 0 [5,2,1,1,1,1,1] => 0 [5,1,1,1,1,1,1,1] => 3 [4,4,4] => 4 [4,4,3,1] => 0 [4,4,2,2] => 2 [4,4,2,1,1] => 0 [4,4,1,1,1,1] => 2 [4,3,3,2] => 0 [4,3,3,1,1] => 0 [4,3,2,2,1] => 0 [4,3,2,1,1,1] => 0 [4,3,1,1,1,1,1] => 0 [4,2,2,2,2] => -2 [4,2,2,2,1,1] => 0 [4,2,2,1,1,1,1] => 0 [4,2,1,1,1,1,1,1] => 0 [4,1,1,1,1,1,1,1,1] => -2 [3,3,3,3] => -3 [3,3,3,2,1] => 0 [3,3,3,1,1,1] => 1 [3,3,2,2,2] => 0 [3,3,2,2,1,1] => 0 [3,3,2,1,1,1,1] => 0 [3,3,1,1,1,1,1,1] => 1 [3,2,2,2,2,1] => 0 [3,2,2,2,1,1,1] => 0 [3,2,2,1,1,1,1,1] => 0 [3,2,1,1,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1,1,1] => 1 [2,2,2,2,2,2] => -2 [2,2,2,2,2,1,1] => 0 [2,2,2,2,1,1,1,1] => 0 [2,2,2,1,1,1,1,1,1] => 0 [2,2,1,1,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1,1,1] => -1 ----------------------------------------------------------------------------- Created: May 03, 2016 at 08:01 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: May 03, 2016 at 11:59 by Martin Rubey