***************************************************************************** * 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: St001383 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The BG-rank of an integer partition. This is the sum of the entries in a checkerboard filling of the Ferrers diagram of the partition, where cells whose sum of coordinates is even have entry $+1$ and the others have entry $-1$. ----------------------------------------------------------------------------- References: [1] Chen, W. Y. C., Ji, K. Q., Wilf, H. S. BG-ranks and 2-cores [[MathSciNet:2274300]] [[arXiv:math/0605474]] ----------------------------------------------------------------------------- Code: def statistic(la): return sum((-1)^i*is_odd(p) for i, p in enumerate(la)) ----------------------------------------------------------------------------- Statistic values: [1] => 1 [2] => 0 [1,1] => 0 [3] => 1 [2,1] => -1 [1,1,1] => 1 [4] => 0 [3,1] => 0 [2,2] => 0 [2,1,1] => 0 [1,1,1,1] => 0 [5] => 1 [4,1] => -1 [3,2] => 1 [3,1,1] => 1 [2,2,1] => 1 [2,1,1,1] => -1 [1,1,1,1,1] => 1 [6] => 0 [5,1] => 0 [4,2] => 0 [4,1,1] => 0 [3,3] => 0 [3,2,1] => 2 [3,1,1,1] => 0 [2,2,2] => 0 [2,2,1,1] => 0 [2,1,1,1,1] => 0 [1,1,1,1,1,1] => 0 [7] => 1 [6,1] => -1 [5,2] => 1 [5,1,1] => 1 [4,3] => -1 [4,2,1] => 1 [4,1,1,1] => -1 [3,3,1] => 1 [3,2,2] => 1 [3,2,1,1] => 1 [3,1,1,1,1] => 1 [2,2,2,1] => -1 [2,2,1,1,1] => 1 [2,1,1,1,1,1] => -1 [1,1,1,1,1,1,1] => 1 [8] => 0 [7,1] => 0 [6,2] => 0 [6,1,1] => 0 [5,3] => 0 [5,2,1] => 2 [5,1,1,1] => 0 [4,4] => 0 [4,3,1] => 0 [4,2,2] => 0 [4,2,1,1] => 0 [4,1,1,1,1] => 0 [3,3,2] => 0 [3,3,1,1] => 0 [3,2,2,1] => 0 [3,2,1,1,1] => 2 [3,1,1,1,1,1] => 0 [2,2,2,2] => 0 [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] => 0 [9] => 1 [8,1] => -1 [7,2] => 1 [7,1,1] => 1 [6,3] => -1 [6,2,1] => 1 [6,1,1,1] => -1 [5,4] => 1 [5,3,1] => 1 [5,2,2] => 1 [5,2,1,1] => 1 [5,1,1,1,1] => 1 [4,4,1] => 1 [4,3,2] => -1 [4,3,1,1] => -1 [4,2,2,1] => -1 [4,2,1,1,1] => 1 [4,1,1,1,1,1] => -1 [3,3,3] => 1 [3,3,2,1] => -1 [3,3,1,1,1] => 1 [3,2,2,2] => 1 [3,2,2,1,1] => 1 [3,2,1,1,1,1] => 1 [3,1,1,1,1,1,1] => 1 [2,2,2,2,1] => 1 [2,2,2,1,1,1] => -1 [2,2,1,1,1,1,1] => 1 [2,1,1,1,1,1,1,1] => -1 [1,1,1,1,1,1,1,1,1] => 1 [10] => 0 [9,1] => 0 [8,2] => 0 [8,1,1] => 0 [7,3] => 0 [7,2,1] => 2 [7,1,1,1] => 0 [6,4] => 0 [6,3,1] => 0 [6,2,2] => 0 [6,2,1,1] => 0 [6,1,1,1,1] => 0 [5,5] => 0 [5,4,1] => 2 [5,3,2] => 0 [5,3,1,1] => 0 [5,2,2,1] => 0 [5,2,1,1,1] => 2 [5,1,1,1,1,1] => 0 [4,4,2] => 0 [4,4,1,1] => 0 [4,3,3] => 0 [4,3,2,1] => -2 [4,3,1,1,1] => 0 [4,2,2,2] => 0 [4,2,2,1,1] => 0 [4,2,1,1,1,1] => 0 [4,1,1,1,1,1,1] => 0 [3,3,3,1] => 0 [3,3,2,2] => 0 [3,3,2,1,1] => 0 [3,3,1,1,1,1] => 0 [3,2,2,2,1] => 2 [3,2,2,1,1,1] => 0 [3,2,1,1,1,1,1] => 2 [3,1,1,1,1,1,1,1] => 0 [2,2,2,2,2] => 0 [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] => 0 [11] => 1 [10,1] => -1 [9,2] => 1 [9,1,1] => 1 [8,3] => -1 [8,2,1] => 1 [8,1,1,1] => -1 [7,4] => 1 [7,3,1] => 1 [7,2,2] => 1 [7,2,1,1] => 1 [7,1,1,1,1] => 1 [6,5] => -1 [6,4,1] => 1 [6,3,2] => -1 [6,3,1,1] => -1 [6,2,2,1] => -1 [6,2,1,1,1] => 1 [6,1,1,1,1,1] => -1 [5,5,1] => 1 [5,4,2] => 1 [5,4,1,1] => 1 [5,3,3] => 1 [5,3,2,1] => -1 [5,3,1,1,1] => 1 [5,2,2,2] => 1 [5,2,2,1,1] => 1 [5,2,1,1,1,1] => 1 [5,1,1,1,1,1,1] => 1 [4,4,3] => 1 [4,4,2,1] => -1 [4,4,1,1,1] => 1 [4,3,3,1] => -1 [4,3,2,2] => -1 [4,3,2,1,1] => -1 [4,3,1,1,1,1] => -1 [4,2,2,2,1] => 1 [4,2,2,1,1,1] => -1 [4,2,1,1,1,1,1] => 1 [4,1,1,1,1,1,1,1] => -1 [3,3,3,2] => 1 [3,3,3,1,1] => 1 [3,3,2,2,1] => 1 [3,3,2,1,1,1] => -1 [3,3,1,1,1,1,1] => 1 [3,2,2,2,2] => 1 [3,2,2,2,1,1] => 1 [3,2,2,1,1,1,1] => 1 [3,2,1,1,1,1,1,1] => 1 [3,1,1,1,1,1,1,1,1] => 1 [2,2,2,2,2,1] => -1 [2,2,2,2,1,1,1] => 1 [2,2,2,1,1,1,1,1] => -1 [2,2,1,1,1,1,1,1,1] => 1 [2,1,1,1,1,1,1,1,1,1] => -1 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 0 [11,1] => 0 [10,2] => 0 [10,1,1] => 0 [9,3] => 0 [9,2,1] => 2 [9,1,1,1] => 0 [8,4] => 0 [8,3,1] => 0 [8,2,2] => 0 [8,2,1,1] => 0 [8,1,1,1,1] => 0 [7,5] => 0 [7,4,1] => 2 [7,3,2] => 0 [7,3,1,1] => 0 [7,2,2,1] => 0 [7,2,1,1,1] => 2 [7,1,1,1,1,1] => 0 [6,6] => 0 [6,5,1] => 0 [6,4,2] => 0 [6,4,1,1] => 0 [6,3,3] => 0 [6,3,2,1] => -2 [6,3,1,1,1] => 0 [6,2,2,2] => 0 [6,2,2,1,1] => 0 [6,2,1,1,1,1] => 0 [6,1,1,1,1,1,1] => 0 [5,5,2] => 0 [5,5,1,1] => 0 [5,4,3] => 2 [5,4,2,1] => 0 [5,4,1,1,1] => 2 [5,3,3,1] => 0 [5,3,2,2] => 0 [5,3,2,1,1] => 0 [5,3,1,1,1,1] => 0 [5,2,2,2,1] => 2 [5,2,2,1,1,1] => 0 [5,2,1,1,1,1,1] => 2 [5,1,1,1,1,1,1,1] => 0 [4,4,4] => 0 [4,4,3,1] => 0 [4,4,2,2] => 0 [4,4,2,1,1] => 0 [4,4,1,1,1,1] => 0 [4,3,3,2] => 0 [4,3,3,1,1] => 0 [4,3,2,2,1] => 0 [4,3,2,1,1,1] => -2 [4,3,1,1,1,1,1] => 0 [4,2,2,2,2] => 0 [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] => 0 [3,3,3,3] => 0 [3,3,3,2,1] => 2 [3,3,3,1,1,1] => 0 [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] => 0 [3,2,2,2,2,1] => 0 [3,2,2,2,1,1,1] => 2 [3,2,2,1,1,1,1,1] => 0 [3,2,1,1,1,1,1,1,1] => 2 [3,1,1,1,1,1,1,1,1,1] => 0 [2,2,2,2,2,2] => 0 [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] => 0 [13] => 1 [12,1] => -1 [11,2] => 1 [11,1,1] => 1 [10,3] => -1 [10,2,1] => 1 [10,1,1,1] => -1 [9,4] => 1 [9,3,1] => 1 [9,2,2] => 1 [9,2,1,1] => 1 [9,1,1,1,1] => 1 [8,5] => -1 [8,4,1] => 1 [8,3,2] => -1 [8,3,1,1] => -1 [8,2,2,1] => -1 [8,2,1,1,1] => 1 [8,1,1,1,1,1] => -1 [7,6] => 1 [7,5,1] => 1 [7,4,2] => 1 [7,4,1,1] => 1 [7,3,3] => 1 [7,3,2,1] => -1 [7,3,1,1,1] => 1 [7,2,2,2] => 1 [7,2,2,1,1] => 1 [7,2,1,1,1,1] => 1 [7,1,1,1,1,1,1] => 1 [6,6,1] => 1 [6,5,2] => -1 [6,5,1,1] => -1 [6,4,3] => 1 [6,4,2,1] => -1 [6,4,1,1,1] => 1 [6,3,3,1] => -1 [6,3,2,2] => -1 [6,3,2,1,1] => -1 [6,3,1,1,1,1] => -1 [6,2,2,2,1] => 1 [6,2,2,1,1,1] => -1 [6,2,1,1,1,1,1] => 1 [6,1,1,1,1,1,1,1] => -1 [5,5,3] => 1 [5,5,2,1] => -1 [5,5,1,1,1] => 1 [5,4,4] => 1 [5,4,3,1] => 1 [5,4,2,2] => 1 [5,4,2,1,1] => 1 [5,4,1,1,1,1] => 1 [5,3,3,2] => 1 [5,3,3,1,1] => 1 [5,3,2,2,1] => 1 [5,3,2,1,1,1] => -1 [5,3,1,1,1,1,1] => 1 [5,2,2,2,2] => 1 [5,2,2,2,1,1] => 1 [5,2,2,1,1,1,1] => 1 [5,2,1,1,1,1,1,1] => 1 [5,1,1,1,1,1,1,1,1] => 1 [4,4,4,1] => -1 [4,4,3,2] => 1 [4,4,3,1,1] => 1 [4,4,2,2,1] => 1 [4,4,2,1,1,1] => -1 [4,4,1,1,1,1,1] => 1 [4,3,3,3] => -1 [4,3,3,2,1] => 1 [4,3,3,1,1,1] => -1 [4,3,2,2,2] => -1 [4,3,2,2,1,1] => -1 [4,3,2,1,1,1,1] => -1 [4,3,1,1,1,1,1,1] => -1 [4,2,2,2,2,1] => -1 [4,2,2,2,1,1,1] => 1 [4,2,2,1,1,1,1,1] => -1 [4,2,1,1,1,1,1,1,1] => 1 [4,1,1,1,1,1,1,1,1,1] => -1 [3,3,3,3,1] => 1 [3,3,3,2,2] => 1 [3,3,3,2,1,1] => 1 [3,3,3,1,1,1,1] => 1 [3,3,2,2,2,1] => -1 [3,3,2,2,1,1,1] => 1 [3,3,2,1,1,1,1,1] => -1 [3,3,1,1,1,1,1,1,1] => 1 [3,2,2,2,2,2] => 1 [3,2,2,2,2,1,1] => 1 [3,2,2,2,1,1,1,1] => 1 [3,2,2,1,1,1,1,1,1] => 1 [3,2,1,1,1,1,1,1,1,1] => 1 [3,1,1,1,1,1,1,1,1,1,1] => 1 [2,2,2,2,2,2,1] => 1 [2,2,2,2,2,1,1,1] => -1 [2,2,2,2,1,1,1,1,1] => 1 [2,2,2,1,1,1,1,1,1,1] => -1 [2,2,1,1,1,1,1,1,1,1,1] => 1 [2,1,1,1,1,1,1,1,1,1,1,1] => -1 [1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [14] => 0 [13,1] => 0 [12,2] => 0 [12,1,1] => 0 [11,3] => 0 [11,2,1] => 2 [11,1,1,1] => 0 [10,4] => 0 [10,3,1] => 0 [10,2,2] => 0 [10,2,1,1] => 0 [10,1,1,1,1] => 0 [9,5] => 0 [9,4,1] => 2 [9,3,2] => 0 [9,3,1,1] => 0 [9,2,2,1] => 0 [9,2,1,1,1] => 2 [9,1,1,1,1,1] => 0 [8,6] => 0 [8,5,1] => 0 [8,4,2] => 0 [8,4,1,1] => 0 [8,3,3] => 0 [8,3,2,1] => -2 [8,3,1,1,1] => 0 [8,2,2,2] => 0 [8,2,2,1,1] => 0 [8,2,1,1,1,1] => 0 [8,1,1,1,1,1,1] => 0 [7,7] => 0 [7,6,1] => 2 [7,5,2] => 0 [7,5,1,1] => 0 [7,4,3] => 2 [7,4,2,1] => 0 [7,4,1,1,1] => 2 [7,3,3,1] => 0 [7,3,2,2] => 0 [7,3,2,1,1] => 0 [7,3,1,1,1,1] => 0 [7,2,2,2,1] => 2 [7,2,2,1,1,1] => 0 [7,2,1,1,1,1,1] => 2 [7,1,1,1,1,1,1,1] => 0 [6,6,2] => 0 [6,6,1,1] => 0 [6,5,3] => 0 [6,5,2,1] => -2 [6,5,1,1,1] => 0 [6,4,4] => 0 [6,4,3,1] => 0 [6,4,2,2] => 0 [6,4,2,1,1] => 0 [6,4,1,1,1,1] => 0 [6,3,3,2] => 0 [6,3,3,1,1] => 0 [6,3,2,2,1] => 0 [6,3,2,1,1,1] => -2 [6,3,1,1,1,1,1] => 0 [6,2,2,2,2] => 0 [6,2,2,2,1,1] => 0 [6,2,2,1,1,1,1] => 0 [6,2,1,1,1,1,1,1] => 0 [6,1,1,1,1,1,1,1,1] => 0 [5,5,4] => 0 [5,5,3,1] => 0 [5,5,2,2] => 0 [5,5,2,1,1] => 0 [5,5,1,1,1,1] => 0 [5,4,4,1] => 0 [5,4,3,2] => 2 [5,4,3,1,1] => 2 [5,4,2,2,1] => 2 [5,4,2,1,1,1] => 0 [5,4,1,1,1,1,1] => 2 [5,3,3,3] => 0 [5,3,3,2,1] => 2 [5,3,3,1,1,1] => 0 [5,3,2,2,2] => 0 [5,3,2,2,1,1] => 0 [5,3,2,1,1,1,1] => 0 [5,3,1,1,1,1,1,1] => 0 [5,2,2,2,2,1] => 0 [5,2,2,2,1,1,1] => 2 [5,2,2,1,1,1,1,1] => 0 [5,2,1,1,1,1,1,1,1] => 2 [5,1,1,1,1,1,1,1,1,1] => 0 [4,4,4,2] => 0 [4,4,4,1,1] => 0 [4,4,3,3] => 0 [4,4,3,2,1] => 2 [4,4,3,1,1,1] => 0 [4,4,2,2,2] => 0 [4,4,2,2,1,1] => 0 [4,4,2,1,1,1,1] => 0 [4,4,1,1,1,1,1,1] => 0 [4,3,3,3,1] => 0 [4,3,3,2,2] => 0 [4,3,3,2,1,1] => 0 [4,3,3,1,1,1,1] => 0 [4,3,2,2,2,1] => -2 [4,3,2,2,1,1,1] => 0 [4,3,2,1,1,1,1,1] => -2 [4,3,1,1,1,1,1,1,1] => 0 [4,2,2,2,2,2] => 0 [4,2,2,2,2,1,1] => 0 [4,2,2,2,1,1,1,1] => 0 [4,2,2,1,1,1,1,1,1] => 0 [4,2,1,1,1,1,1,1,1,1] => 0 [4,1,1,1,1,1,1,1,1,1,1] => 0 [3,3,3,3,2] => 0 [3,3,3,3,1,1] => 0 [3,3,3,2,2,1] => 0 [3,3,3,2,1,1,1] => 2 [3,3,3,1,1,1,1,1] => 0 [3,3,2,2,2,2] => 0 [3,3,2,2,2,1,1] => 0 [3,3,2,2,1,1,1,1] => 0 [3,3,2,1,1,1,1,1,1] => 0 [3,3,1,1,1,1,1,1,1,1] => 0 [3,2,2,2,2,2,1] => 2 [3,2,2,2,2,1,1,1] => 0 [3,2,2,2,1,1,1,1,1] => 2 [3,2,2,1,1,1,1,1,1,1] => 0 [3,2,1,1,1,1,1,1,1,1,1] => 2 [3,1,1,1,1,1,1,1,1,1,1,1] => 0 [2,2,2,2,2,2,2] => 0 [2,2,2,2,2,2,1,1] => 0 [2,2,2,2,2,1,1,1,1] => 0 [2,2,2,2,1,1,1,1,1,1] => 0 [2,2,2,1,1,1,1,1,1,1,1] => 0 [2,2,1,1,1,1,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0 [15] => 1 [14,1] => -1 [13,2] => 1 [13,1,1] => 1 [12,3] => -1 [12,2,1] => 1 [12,1,1,1] => -1 [11,4] => 1 [11,3,1] => 1 [11,2,2] => 1 [11,2,1,1] => 1 [11,1,1,1,1] => 1 [10,5] => -1 [10,4,1] => 1 [10,3,2] => -1 [10,3,1,1] => -1 [10,2,2,1] => -1 [10,2,1,1,1] => 1 [10,1,1,1,1,1] => -1 [9,6] => 1 [9,5,1] => 1 [9,4,2] => 1 [9,4,1,1] => 1 [9,3,3] => 1 [9,3,2,1] => -1 [9,3,1,1,1] => 1 [9,2,2,2] => 1 [9,2,2,1,1] => 1 [9,2,1,1,1,1] => 1 [9,1,1,1,1,1,1] => 1 [8,7] => -1 [8,6,1] => 1 [8,5,2] => -1 [8,5,1,1] => -1 [8,4,3] => 1 [8,4,2,1] => -1 [8,4,1,1,1] => 1 [8,3,3,1] => -1 [8,3,2,2] => -1 [8,3,2,1,1] => -1 [8,3,1,1,1,1] => -1 [8,2,2,2,1] => 1 [8,2,2,1,1,1] => -1 [8,2,1,1,1,1,1] => 1 [8,1,1,1,1,1,1,1] => -1 [7,7,1] => 1 [7,6,2] => 1 [7,6,1,1] => 1 [7,5,3] => 1 [7,5,2,1] => -1 [7,5,1,1,1] => 1 [7,4,4] => 1 [7,4,3,1] => 1 [7,4,2,2] => 1 [7,4,2,1,1] => 1 [7,4,1,1,1,1] => 1 [7,3,3,2] => 1 [7,3,3,1,1] => 1 [7,3,2,2,1] => 1 [7,3,2,1,1,1] => -1 [7,3,1,1,1,1,1] => 1 [7,2,2,2,2] => 1 [7,2,2,2,1,1] => 1 [7,2,2,1,1,1,1] => 1 [7,2,1,1,1,1,1,1] => 1 [7,1,1,1,1,1,1,1,1] => 1 [6,6,3] => 1 [6,6,2,1] => -1 [6,6,1,1,1] => 1 [6,5,4] => -1 [6,5,3,1] => -1 [6,5,2,2] => -1 [6,5,2,1,1] => -1 [6,5,1,1,1,1] => -1 [6,4,4,1] => -1 [6,4,3,2] => 1 [6,4,3,1,1] => 1 [6,4,2,2,1] => 1 [6,4,2,1,1,1] => -1 [6,4,1,1,1,1,1] => 1 [6,3,3,3] => -1 [6,3,3,2,1] => 1 [6,3,3,1,1,1] => -1 [6,3,2,2,2] => -1 [6,3,2,2,1,1] => -1 [6,3,2,1,1,1,1] => -1 [6,3,1,1,1,1,1,1] => -1 [6,2,2,2,2,1] => -1 [6,2,2,2,1,1,1] => 1 [6,2,2,1,1,1,1,1] => -1 [6,2,1,1,1,1,1,1,1] => 1 [6,1,1,1,1,1,1,1,1,1] => -1 [5,5,5] => 1 [5,5,4,1] => -1 [5,5,3,2] => 1 [5,5,3,1,1] => 1 [5,5,2,2,1] => 1 [5,5,2,1,1,1] => -1 [5,5,1,1,1,1,1] => 1 [5,4,4,2] => 1 [5,4,4,1,1] => 1 [5,4,3,3] => 1 [5,4,3,2,1] => 3 [5,4,3,1,1,1] => 1 [5,4,2,2,2] => 1 [5,4,2,2,1,1] => 1 [5,4,2,1,1,1,1] => 1 [5,4,1,1,1,1,1,1] => 1 [5,3,3,3,1] => 1 [5,3,3,2,2] => 1 [5,3,3,2,1,1] => 1 [5,3,3,1,1,1,1] => 1 [5,3,2,2,2,1] => -1 [5,3,2,2,1,1,1] => 1 [5,3,2,1,1,1,1,1] => -1 [5,3,1,1,1,1,1,1,1] => 1 [5,2,2,2,2,2] => 1 [5,2,2,2,2,1,1] => 1 [5,2,2,2,1,1,1,1] => 1 [5,2,2,1,1,1,1,1,1] => 1 [5,2,1,1,1,1,1,1,1,1] => 1 [5,1,1,1,1,1,1,1,1,1,1] => 1 [4,4,4,3] => -1 [4,4,4,2,1] => 1 [4,4,4,1,1,1] => -1 [4,4,3,3,1] => 1 [4,4,3,2,2] => 1 [4,4,3,2,1,1] => 1 [4,4,3,1,1,1,1] => 1 [4,4,2,2,2,1] => -1 [4,4,2,2,1,1,1] => 1 [4,4,2,1,1,1,1,1] => -1 [4,4,1,1,1,1,1,1,1] => 1 [4,3,3,3,2] => -1 [4,3,3,3,1,1] => -1 [4,3,3,2,2,1] => -1 [4,3,3,2,1,1,1] => 1 [4,3,3,1,1,1,1,1] => -1 [4,3,2,2,2,2] => -1 [4,3,2,2,2,1,1] => -1 [4,3,2,2,1,1,1,1] => -1 [4,3,2,1,1,1,1,1,1] => -1 [4,3,1,1,1,1,1,1,1,1] => -1 [4,2,2,2,2,2,1] => 1 [4,2,2,2,2,1,1,1] => -1 [4,2,2,2,1,1,1,1,1] => 1 [4,2,2,1,1,1,1,1,1,1] => -1 [4,2,1,1,1,1,1,1,1,1,1] => 1 [4,1,1,1,1,1,1,1,1,1,1,1] => -1 [3,3,3,3,3] => 1 [3,3,3,3,2,1] => -1 [3,3,3,3,1,1,1] => 1 [3,3,3,2,2,2] => 1 [3,3,3,2,2,1,1] => 1 [3,3,3,2,1,1,1,1] => 1 [3,3,3,1,1,1,1,1,1] => 1 [3,3,2,2,2,2,1] => 1 [3,3,2,2,2,1,1,1] => -1 [3,3,2,2,1,1,1,1,1] => 1 [3,3,2,1,1,1,1,1,1,1] => -1 [3,3,1,1,1,1,1,1,1,1,1] => 1 [3,2,2,2,2,2,2] => 1 [3,2,2,2,2,2,1,1] => 1 [3,2,2,2,2,1,1,1,1] => 1 [3,2,2,2,1,1,1,1,1,1] => 1 [3,2,2,1,1,1,1,1,1,1,1] => 1 [3,2,1,1,1,1,1,1,1,1,1,1] => 1 [3,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [2,2,2,2,2,2,2,1] => -1 [2,2,2,2,2,2,1,1,1] => 1 [2,2,2,2,2,1,1,1,1,1] => -1 [2,2,2,2,1,1,1,1,1,1,1] => 1 [2,2,2,1,1,1,1,1,1,1,1,1] => -1 [2,2,1,1,1,1,1,1,1,1,1,1,1] => 1 [2,1,1,1,1,1,1,1,1,1,1,1,1,1] => -1 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [16] => 0 [15,1] => 0 [14,2] => 0 [14,1,1] => 0 [13,3] => 0 [13,2,1] => 2 [13,1,1,1] => 0 [12,4] => 0 [12,3,1] => 0 [12,2,2] => 0 [12,2,1,1] => 0 [12,1,1,1,1] => 0 [11,5] => 0 [11,4,1] => 2 [11,3,2] => 0 [11,3,1,1] => 0 [11,2,2,1] => 0 [11,2,1,1,1] => 2 [11,1,1,1,1,1] => 0 [10,6] => 0 [10,5,1] => 0 [10,4,2] => 0 [10,4,1,1] => 0 [10,3,3] => 0 [10,3,2,1] => -2 [10,3,1,1,1] => 0 [10,2,2,2] => 0 [10,2,2,1,1] => 0 [10,2,1,1,1,1] => 0 [10,1,1,1,1,1,1] => 0 [9,7] => 0 [9,6,1] => 2 [9,5,2] => 0 [9,5,1,1] => 0 [9,4,3] => 2 [9,4,2,1] => 0 [9,4,1,1,1] => 2 [9,3,3,1] => 0 [9,3,2,2] => 0 [9,3,2,1,1] => 0 [9,3,1,1,1,1] => 0 [9,2,2,2,1] => 2 [9,2,2,1,1,1] => 0 [9,2,1,1,1,1,1] => 2 [9,1,1,1,1,1,1,1] => 0 [8,8] => 0 [8,7,1] => 0 [8,6,2] => 0 [8,6,1,1] => 0 [8,5,3] => 0 [8,5,2,1] => -2 [8,5,1,1,1] => 0 [8,4,4] => 0 [8,4,3,1] => 0 [8,4,2,2] => 0 [8,4,2,1,1] => 0 [8,4,1,1,1,1] => 0 [8,3,3,2] => 0 [8,3,3,1,1] => 0 [8,3,2,2,1] => 0 [8,3,2,1,1,1] => -2 [8,3,1,1,1,1,1] => 0 [8,2,2,2,2] => 0 [8,2,2,2,1,1] => 0 [8,2,2,1,1,1,1] => 0 [8,2,1,1,1,1,1,1] => 0 [8,1,1,1,1,1,1,1,1] => 0 [7,7,2] => 0 [7,7,1,1] => 0 [7,6,3] => 2 [7,6,2,1] => 0 [7,6,1,1,1] => 2 [7,5,4] => 0 [7,5,3,1] => 0 [7,5,2,2] => 0 [7,5,2,1,1] => 0 [7,5,1,1,1,1] => 0 [7,4,4,1] => 0 [7,4,3,2] => 2 [7,4,3,1,1] => 2 [7,4,2,2,1] => 2 [7,4,2,1,1,1] => 0 [7,4,1,1,1,1,1] => 2 [7,3,3,3] => 0 [7,3,3,2,1] => 2 [7,3,3,1,1,1] => 0 [7,3,2,2,2] => 0 [7,3,2,2,1,1] => 0 [7,3,2,1,1,1,1] => 0 [7,3,1,1,1,1,1,1] => 0 [7,2,2,2,2,1] => 0 [7,2,2,2,1,1,1] => 2 [7,2,2,1,1,1,1,1] => 0 [7,2,1,1,1,1,1,1,1] => 2 [7,1,1,1,1,1,1,1,1,1] => 0 [6,6,4] => 0 [6,6,3,1] => 0 [6,6,2,2] => 0 [6,6,2,1,1] => 0 [6,6,1,1,1,1] => 0 [6,5,5] => 0 [6,5,4,1] => -2 [6,5,3,2] => 0 [6,5,3,1,1] => 0 [6,5,2,2,1] => 0 [6,5,2,1,1,1] => -2 [6,5,1,1,1,1,1] => 0 [6,4,4,2] => 0 [6,4,4,1,1] => 0 [6,4,3,3] => 0 [6,4,3,2,1] => 2 [6,4,3,1,1,1] => 0 [6,4,2,2,2] => 0 [6,4,2,2,1,1] => 0 [6,4,2,1,1,1,1] => 0 [6,4,1,1,1,1,1,1] => 0 [6,3,3,3,1] => 0 [6,3,3,2,2] => 0 [6,3,3,2,1,1] => 0 [6,3,3,1,1,1,1] => 0 [6,3,2,2,2,1] => -2 [6,3,2,2,1,1,1] => 0 [6,3,2,1,1,1,1,1] => -2 [6,3,1,1,1,1,1,1,1] => 0 [6,2,2,2,2,2] => 0 [6,2,2,2,2,1,1] => 0 [6,2,2,2,1,1,1,1] => 0 [6,2,2,1,1,1,1,1,1] => 0 [6,2,1,1,1,1,1,1,1,1] => 0 [6,1,1,1,1,1,1,1,1,1,1] => 0 [5,5,5,1] => 0 [5,5,4,2] => 0 [5,5,4,1,1] => 0 [5,5,3,3] => 0 [5,5,3,2,1] => 2 [5,5,3,1,1,1] => 0 [5,5,2,2,2] => 0 [5,5,2,2,1,1] => 0 [5,5,2,1,1,1,1] => 0 [5,5,1,1,1,1,1,1] => 0 [5,4,4,3] => 0 [5,4,4,2,1] => 2 [5,4,4,1,1,1] => 0 [5,4,3,3,1] => 2 [5,4,3,2,2] => 2 [5,4,3,2,1,1] => 2 [5,4,3,1,1,1,1] => 2 [5,4,2,2,2,1] => 0 [5,4,2,2,1,1,1] => 2 [5,4,2,1,1,1,1,1] => 0 [5,4,1,1,1,1,1,1,1] => 2 [5,3,3,3,2] => 0 [5,3,3,3,1,1] => 0 [5,3,3,2,2,1] => 0 [5,3,3,2,1,1,1] => 2 [5,3,3,1,1,1,1,1] => 0 [5,3,2,2,2,2] => 0 [5,3,2,2,2,1,1] => 0 [5,3,2,2,1,1,1,1] => 0 [5,3,2,1,1,1,1,1,1] => 0 [5,3,1,1,1,1,1,1,1,1] => 0 [5,2,2,2,2,2,1] => 2 [5,2,2,2,2,1,1,1] => 0 [5,2,2,2,1,1,1,1,1] => 2 [5,2,2,1,1,1,1,1,1,1] => 0 [5,2,1,1,1,1,1,1,1,1,1] => 2 [5,1,1,1,1,1,1,1,1,1,1,1] => 0 [4,4,4,4] => 0 [4,4,4,3,1] => 0 [4,4,4,2,2] => 0 [4,4,4,2,1,1] => 0 [4,4,4,1,1,1,1] => 0 [4,4,3,3,2] => 0 [4,4,3,3,1,1] => 0 [4,4,3,2,2,1] => 0 [4,4,3,2,1,1,1] => 2 [4,4,3,1,1,1,1,1] => 0 [4,4,2,2,2,2] => 0 [4,4,2,2,2,1,1] => 0 [4,4,2,2,1,1,1,1] => 0 [4,4,2,1,1,1,1,1,1] => 0 [4,4,1,1,1,1,1,1,1,1] => 0 [4,3,3,3,3] => 0 [4,3,3,3,2,1] => -2 [4,3,3,3,1,1,1] => 0 [4,3,3,2,2,2] => 0 [4,3,3,2,2,1,1] => 0 [4,3,3,2,1,1,1,1] => 0 [4,3,3,1,1,1,1,1,1] => 0 [4,3,2,2,2,2,1] => 0 [4,3,2,2,2,1,1,1] => -2 [4,3,2,2,1,1,1,1,1] => 0 [4,3,2,1,1,1,1,1,1,1] => -2 [4,3,1,1,1,1,1,1,1,1,1] => 0 [4,2,2,2,2,2,2] => 0 [4,2,2,2,2,2,1,1] => 0 [4,2,2,2,2,1,1,1,1] => 0 [4,2,2,2,1,1,1,1,1,1] => 0 [4,2,2,1,1,1,1,1,1,1,1] => 0 [4,2,1,1,1,1,1,1,1,1,1,1] => 0 [4,1,1,1,1,1,1,1,1,1,1,1,1] => 0 [3,3,3,3,3,1] => 0 [3,3,3,3,2,2] => 0 [3,3,3,3,2,1,1] => 0 [3,3,3,3,1,1,1,1] => 0 [3,3,3,2,2,2,1] => 2 [3,3,3,2,2,1,1,1] => 0 [3,3,3,2,1,1,1,1,1] => 2 [3,3,3,1,1,1,1,1,1,1] => 0 [3,3,2,2,2,2,2] => 0 [3,3,2,2,2,2,1,1] => 0 [3,3,2,2,2,1,1,1,1] => 0 [3,3,2,2,1,1,1,1,1,1] => 0 [3,3,2,1,1,1,1,1,1,1,1] => 0 [3,3,1,1,1,1,1,1,1,1,1,1] => 0 [3,2,2,2,2,2,2,1] => 0 [3,2,2,2,2,2,1,1,1] => 2 [3,2,2,2,2,1,1,1,1,1] => 0 [3,2,2,2,1,1,1,1,1,1,1] => 2 [3,2,2,1,1,1,1,1,1,1,1,1] => 0 [3,2,1,1,1,1,1,1,1,1,1,1,1] => 2 [3,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0 [2,2,2,2,2,2,2,2] => 0 [2,2,2,2,2,2,2,1,1] => 0 [2,2,2,2,2,2,1,1,1,1] => 0 [2,2,2,2,2,1,1,1,1,1,1] => 0 [2,2,2,2,1,1,1,1,1,1,1,1] => 0 [2,2,2,1,1,1,1,1,1,1,1,1,1] => 0 [2,2,1,1,1,1,1,1,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0 [17] => 1 [16,1] => -1 [15,2] => 1 [15,1,1] => 1 [14,3] => -1 [14,2,1] => 1 [14,1,1,1] => -1 [13,4] => 1 [13,3,1] => 1 [13,2,2] => 1 [13,2,1,1] => 1 [13,1,1,1,1] => 1 [12,5] => -1 [12,4,1] => 1 [12,3,2] => -1 [12,3,1,1] => -1 [12,2,2,1] => -1 [12,2,1,1,1] => 1 [12,1,1,1,1,1] => -1 [11,6] => 1 [11,5,1] => 1 [11,4,2] => 1 [11,4,1,1] => 1 [11,3,3] => 1 [11,3,2,1] => -1 [11,3,1,1,1] => 1 [11,2,2,2] => 1 [11,2,2,1,1] => 1 [11,2,1,1,1,1] => 1 [11,1,1,1,1,1,1] => 1 [10,7] => -1 [10,6,1] => 1 [10,5,2] => -1 [10,5,1,1] => -1 [10,4,3] => 1 [10,4,2,1] => -1 [10,4,1,1,1] => 1 [10,3,3,1] => -1 [10,3,2,2] => -1 [10,3,2,1,1] => -1 [10,3,1,1,1,1] => -1 [10,2,2,2,1] => 1 [10,2,2,1,1,1] => -1 [10,2,1,1,1,1,1] => 1 [10,1,1,1,1,1,1,1] => -1 [9,8] => 1 [9,7,1] => 1 [9,6,2] => 1 [9,6,1,1] => 1 [9,5,3] => 1 [9,5,2,1] => -1 [9,5,1,1,1] => 1 [9,4,4] => 1 [9,4,3,1] => 1 [9,4,2,2] => 1 [9,4,2,1,1] => 1 [9,4,1,1,1,1] => 1 [9,3,3,2] => 1 [9,3,3,1,1] => 1 [9,3,2,2,1] => 1 [9,3,2,1,1,1] => -1 [9,3,1,1,1,1,1] => 1 [9,2,2,2,2] => 1 [9,2,2,2,1,1] => 1 [9,2,2,1,1,1,1] => 1 [9,2,1,1,1,1,1,1] => 1 [9,1,1,1,1,1,1,1,1] => 1 [8,8,1] => 1 [8,7,2] => -1 [8,7,1,1] => -1 [8,6,3] => 1 [8,6,2,1] => -1 [8,6,1,1,1] => 1 [8,5,4] => -1 [8,5,3,1] => -1 [8,5,2,2] => -1 [8,5,2,1,1] => -1 [8,5,1,1,1,1] => -1 [8,4,4,1] => -1 [8,4,3,2] => 1 [8,4,3,1,1] => 1 [8,4,2,2,1] => 1 [8,4,2,1,1,1] => -1 [8,4,1,1,1,1,1] => 1 [8,3,3,3] => -1 [8,3,3,2,1] => 1 [8,3,3,1,1,1] => -1 [8,3,2,2,2] => -1 [8,3,2,2,1,1] => -1 [8,3,2,1,1,1,1] => -1 [8,3,1,1,1,1,1,1] => -1 [8,2,2,2,2,1] => -1 [8,2,2,2,1,1,1] => 1 [8,2,2,1,1,1,1,1] => -1 [8,2,1,1,1,1,1,1,1] => 1 [8,1,1,1,1,1,1,1,1,1] => -1 [7,7,3] => 1 [7,7,2,1] => -1 [7,7,1,1,1] => 1 [7,6,4] => 1 [7,6,3,1] => 1 [7,6,2,2] => 1 [7,6,2,1,1] => 1 [7,6,1,1,1,1] => 1 [7,5,5] => 1 [7,5,4,1] => -1 [7,5,3,2] => 1 [7,5,3,1,1] => 1 [7,5,2,2,1] => 1 [7,5,2,1,1,1] => -1 [7,5,1,1,1,1,1] => 1 [7,4,4,2] => 1 [7,4,4,1,1] => 1 [7,4,3,3] => 1 [7,4,3,2,1] => 3 [7,4,3,1,1,1] => 1 [7,4,2,2,2] => 1 [7,4,2,2,1,1] => 1 [7,4,2,1,1,1,1] => 1 [7,4,1,1,1,1,1,1] => 1 [7,3,3,3,1] => 1 [7,3,3,2,2] => 1 [7,3,3,2,1,1] => 1 [7,3,3,1,1,1,1] => 1 [7,3,2,2,2,1] => -1 [7,3,2,2,1,1,1] => 1 [7,3,2,1,1,1,1,1] => -1 [7,3,1,1,1,1,1,1,1] => 1 [7,2,2,2,2,2] => 1 [7,2,2,2,2,1,1] => 1 [7,2,2,2,1,1,1,1] => 1 [7,2,2,1,1,1,1,1,1] => 1 [7,2,1,1,1,1,1,1,1,1] => 1 [7,1,1,1,1,1,1,1,1,1,1] => 1 [6,6,5] => 1 [6,6,4,1] => -1 [6,6,3,2] => 1 [6,6,3,1,1] => 1 [6,6,2,2,1] => 1 [6,6,2,1,1,1] => -1 [6,6,1,1,1,1,1] => 1 [6,5,5,1] => -1 [6,5,4,2] => -1 [6,5,4,1,1] => -1 [6,5,3,3] => -1 [6,5,3,2,1] => 1 [6,5,3,1,1,1] => -1 [6,5,2,2,2] => -1 [6,5,2,2,1,1] => -1 [6,5,2,1,1,1,1] => -1 [6,5,1,1,1,1,1,1] => -1 [6,4,4,3] => -1 [6,4,4,2,1] => 1 [6,4,4,1,1,1] => -1 [6,4,3,3,1] => 1 [6,4,3,2,2] => 1 [6,4,3,2,1,1] => 1 [6,4,3,1,1,1,1] => 1 [6,4,2,2,2,1] => -1 [6,4,2,2,1,1,1] => 1 [6,4,2,1,1,1,1,1] => -1 [6,4,1,1,1,1,1,1,1] => 1 [6,3,3,3,2] => -1 [6,3,3,3,1,1] => -1 [6,3,3,2,2,1] => -1 [6,3,3,2,1,1,1] => 1 [6,3,3,1,1,1,1,1] => -1 [6,3,2,2,2,2] => -1 [6,3,2,2,2,1,1] => -1 [6,3,2,2,1,1,1,1] => -1 [6,3,2,1,1,1,1,1,1] => -1 [6,3,1,1,1,1,1,1,1,1] => -1 [6,2,2,2,2,2,1] => 1 [6,2,2,2,2,1,1,1] => -1 [6,2,2,2,1,1,1,1,1] => 1 [6,2,2,1,1,1,1,1,1,1] => -1 [6,2,1,1,1,1,1,1,1,1,1] => 1 [6,1,1,1,1,1,1,1,1,1,1,1] => -1 [5,5,5,2] => 1 [5,5,5,1,1] => 1 [5,5,4,3] => -1 [5,5,4,2,1] => 1 [5,5,4,1,1,1] => -1 [5,5,3,3,1] => 1 [5,5,3,2,2] => 1 [5,5,3,2,1,1] => 1 [5,5,3,1,1,1,1] => 1 [5,5,2,2,2,1] => -1 [5,5,2,2,1,1,1] => 1 [5,5,2,1,1,1,1,1] => -1 [5,5,1,1,1,1,1,1,1] => 1 [5,4,4,4] => 1 [5,4,4,3,1] => 1 [5,4,4,2,2] => 1 [5,4,4,2,1,1] => 1 [5,4,4,1,1,1,1] => 1 [5,4,3,3,2] => 1 [5,4,3,3,1,1] => 1 [5,4,3,2,2,1] => 1 [5,4,3,2,1,1,1] => 3 [5,4,3,1,1,1,1,1] => 1 [5,4,2,2,2,2] => 1 [5,4,2,2,2,1,1] => 1 [5,4,2,2,1,1,1,1] => 1 [5,4,2,1,1,1,1,1,1] => 1 [5,4,1,1,1,1,1,1,1,1] => 1 [5,3,3,3,3] => 1 [5,3,3,3,2,1] => -1 [5,3,3,3,1,1,1] => 1 [5,3,3,2,2,2] => 1 [5,3,3,2,2,1,1] => 1 [5,3,3,2,1,1,1,1] => 1 [5,3,3,1,1,1,1,1,1] => 1 [5,3,2,2,2,2,1] => 1 [5,3,2,2,2,1,1,1] => -1 [5,3,2,2,1,1,1,1,1] => 1 [5,3,2,1,1,1,1,1,1,1] => -1 [5,3,1,1,1,1,1,1,1,1,1] => 1 [5,2,2,2,2,2,2] => 1 [5,2,2,2,2,2,1,1] => 1 [5,2,2,2,2,1,1,1,1] => 1 [5,2,2,2,1,1,1,1,1,1] => 1 [5,2,2,1,1,1,1,1,1,1,1] => 1 [5,2,1,1,1,1,1,1,1,1,1,1] => 1 [5,1,1,1,1,1,1,1,1,1,1,1,1] => 1 [4,4,4,4,1] => 1 [4,4,4,3,2] => -1 [4,4,4,3,1,1] => -1 [4,4,4,2,2,1] => -1 [4,4,4,2,1,1,1] => 1 [4,4,4,1,1,1,1,1] => -1 [4,4,3,3,3] => 1 [4,4,3,3,2,1] => -1 [4,4,3,3,1,1,1] => 1 [4,4,3,2,2,2] => 1 [4,4,3,2,2,1,1] => 1 [4,4,3,2,1,1,1,1] => 1 [4,4,3,1,1,1,1,1,1] => 1 [4,4,2,2,2,2,1] => 1 [4,4,2,2,2,1,1,1] => -1 [4,4,2,2,1,1,1,1,1] => 1 [4,4,2,1,1,1,1,1,1,1] => -1 [4,4,1,1,1,1,1,1,1,1,1] => 1 [4,3,3,3,3,1] => -1 [4,3,3,3,2,2] => -1 [4,3,3,3,2,1,1] => -1 [4,3,3,3,1,1,1,1] => -1 [4,3,3,2,2,2,1] => 1 [4,3,3,2,2,1,1,1] => -1 [4,3,3,2,1,1,1,1,1] => 1 [4,3,3,1,1,1,1,1,1,1] => -1 [4,3,2,2,2,2,2] => -1 [4,3,2,2,2,2,1,1] => -1 [4,3,2,2,2,1,1,1,1] => -1 [4,3,2,2,1,1,1,1,1,1] => -1 [4,3,2,1,1,1,1,1,1,1,1] => -1 [4,3,1,1,1,1,1,1,1,1,1,1] => -1 [4,2,2,2,2,2,2,1] => -1 [4,2,2,2,2,2,1,1,1] => 1 [4,2,2,2,2,1,1,1,1,1] => -1 [4,2,2,2,1,1,1,1,1,1,1] => 1 [4,2,2,1,1,1,1,1,1,1,1,1] => -1 [4,2,1,1,1,1,1,1,1,1,1,1,1] => 1 [4,1,1,1,1,1,1,1,1,1,1,1,1,1] => -1 [3,3,3,3,3,2] => 1 [3,3,3,3,3,1,1] => 1 [3,3,3,3,2,2,1] => 1 [3,3,3,3,2,1,1,1] => -1 [3,3,3,3,1,1,1,1,1] => 1 [3,3,3,2,2,2,2] => 1 [3,3,3,2,2,2,1,1] => 1 [3,3,3,2,2,1,1,1,1] => 1 [3,3,3,2,1,1,1,1,1,1] => 1 [3,3,3,1,1,1,1,1,1,1,1] => 1 [3,3,2,2,2,2,2,1] => -1 [3,3,2,2,2,2,1,1,1] => 1 [3,3,2,2,2,1,1,1,1,1] => -1 [3,3,2,2,1,1,1,1,1,1,1] => 1 [3,3,2,1,1,1,1,1,1,1,1,1] => -1 [3,3,1,1,1,1,1,1,1,1,1,1,1] => 1 [3,2,2,2,2,2,2,2] => 1 [3,2,2,2,2,2,2,1,1] => 1 [3,2,2,2,2,2,1,1,1,1] => 1 [3,2,2,2,2,1,1,1,1,1,1] => 1 [3,2,2,2,1,1,1,1,1,1,1,1] => 1 [3,2,2,1,1,1,1,1,1,1,1,1,1] => 1 ----------------------------------------------------------------------------- Created: Apr 19, 2019 at 18:48 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Apr 19, 2019 at 18:48 by Martin Rubey