***************************************************************************** * 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: St001392 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. ----------------------------------------------------------------------------- References: [1] Chern, S. Partitions and the maximal excludant [[arXiv:1905.06304]] ----------------------------------------------------------------------------- Code: def statistic(p): return next(i for i in range(p[0]-1,-1,-1) if i not in p) ----------------------------------------------------------------------------- Statistic values: [1] => 0 [2] => 1 [1,1] => 0 [3] => 2 [2,1] => 0 [1,1,1] => 0 [4] => 3 [3,1] => 2 [2,2] => 1 [2,1,1] => 0 [1,1,1,1] => 0 [5] => 4 [4,1] => 3 [3,2] => 1 [3,1,1] => 2 [2,2,1] => 0 [2,1,1,1] => 0 [1,1,1,1,1] => 0 [6] => 5 [5,1] => 4 [4,2] => 3 [4,1,1] => 3 [3,3] => 2 [3,2,1] => 0 [3,1,1,1] => 2 [2,2,2] => 1 [2,2,1,1] => 0 [2,1,1,1,1] => 0 [1,1,1,1,1,1] => 0 [7] => 6 [6,1] => 5 [5,2] => 4 [5,1,1] => 4 [4,3] => 2 [4,2,1] => 3 [4,1,1,1] => 3 [3,3,1] => 2 [3,2,2] => 1 [3,2,1,1] => 0 [3,1,1,1,1] => 2 [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] => 0 [8] => 7 [7,1] => 6 [6,2] => 5 [6,1,1] => 5 [5,3] => 4 [5,2,1] => 4 [5,1,1,1] => 4 [4,4] => 3 [4,3,1] => 2 [4,2,2] => 3 [4,2,1,1] => 3 [4,1,1,1,1] => 3 [3,3,2] => 1 [3,3,1,1] => 2 [3,2,2,1] => 0 [3,2,1,1,1] => 0 [3,1,1,1,1,1] => 2 [2,2,2,2] => 1 [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] => 8 [8,1] => 7 [7,2] => 6 [7,1,1] => 6 [6,3] => 5 [6,2,1] => 5 [6,1,1,1] => 5 [5,4] => 3 [5,3,1] => 4 [5,2,2] => 4 [5,2,1,1] => 4 [5,1,1,1,1] => 4 [4,4,1] => 3 [4,3,2] => 1 [4,3,1,1] => 2 [4,2,2,1] => 3 [4,2,1,1,1] => 3 [4,1,1,1,1,1] => 3 [3,3,3] => 2 [3,3,2,1] => 0 [3,3,1,1,1] => 2 [3,2,2,2] => 1 [3,2,2,1,1] => 0 [3,2,1,1,1,1] => 0 [3,1,1,1,1,1,1] => 2 [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] => 0 [10] => 9 [9,1] => 8 [8,2] => 7 [8,1,1] => 7 [7,3] => 6 [7,2,1] => 6 [7,1,1,1] => 6 [6,4] => 5 [6,3,1] => 5 [6,2,2] => 5 [6,2,1,1] => 5 [6,1,1,1,1] => 5 [5,5] => 4 [5,4,1] => 3 [5,3,2] => 4 [5,3,1,1] => 4 [5,2,2,1] => 4 [5,2,1,1,1] => 4 [5,1,1,1,1,1] => 4 [4,4,2] => 3 [4,4,1,1] => 3 [4,3,3] => 2 [4,3,2,1] => 0 [4,3,1,1,1] => 2 [4,2,2,2] => 3 [4,2,2,1,1] => 3 [4,2,1,1,1,1] => 3 [4,1,1,1,1,1,1] => 3 [3,3,3,1] => 2 [3,3,2,2] => 1 [3,3,2,1,1] => 0 [3,3,1,1,1,1] => 2 [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] => 2 [2,2,2,2,2] => 1 [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] => 10 [10,1] => 9 [9,2] => 8 [9,1,1] => 8 [8,3] => 7 [8,2,1] => 7 [8,1,1,1] => 7 [7,4] => 6 [7,3,1] => 6 [7,2,2] => 6 [7,2,1,1] => 6 [7,1,1,1,1] => 6 [6,5] => 4 [6,4,1] => 5 [6,3,2] => 5 [6,3,1,1] => 5 [6,2,2,1] => 5 [6,2,1,1,1] => 5 [6,1,1,1,1,1] => 5 [5,5,1] => 4 [5,4,2] => 3 [5,4,1,1] => 3 [5,3,3] => 4 [5,3,2,1] => 4 [5,3,1,1,1] => 4 [5,2,2,2] => 4 [5,2,2,1,1] => 4 [5,2,1,1,1,1] => 4 [5,1,1,1,1,1,1] => 4 [4,4,3] => 2 [4,4,2,1] => 3 [4,4,1,1,1] => 3 [4,3,3,1] => 2 [4,3,2,2] => 1 [4,3,2,1,1] => 0 [4,3,1,1,1,1] => 2 [4,2,2,2,1] => 3 [4,2,2,1,1,1] => 3 [4,2,1,1,1,1,1] => 3 [4,1,1,1,1,1,1,1] => 3 [3,3,3,2] => 1 [3,3,3,1,1] => 2 [3,3,2,2,1] => 0 [3,3,2,1,1,1] => 0 [3,3,1,1,1,1,1] => 2 [3,2,2,2,2] => 1 [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] => 2 [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] => 0 [12] => 11 [11,1] => 10 [10,2] => 9 [10,1,1] => 9 [9,3] => 8 [9,2,1] => 8 [9,1,1,1] => 8 [8,4] => 7 [8,3,1] => 7 [8,2,2] => 7 [8,2,1,1] => 7 [8,1,1,1,1] => 7 [7,5] => 6 [7,4,1] => 6 [7,3,2] => 6 [7,3,1,1] => 6 [7,2,2,1] => 6 [7,2,1,1,1] => 6 [7,1,1,1,1,1] => 6 [6,6] => 5 [6,5,1] => 4 [6,4,2] => 5 [6,4,1,1] => 5 [6,3,3] => 5 [6,3,2,1] => 5 [6,3,1,1,1] => 5 [6,2,2,2] => 5 [6,2,2,1,1] => 5 [6,2,1,1,1,1] => 5 [6,1,1,1,1,1,1] => 5 [5,5,2] => 4 [5,5,1,1] => 4 [5,4,3] => 2 [5,4,2,1] => 3 [5,4,1,1,1] => 3 [5,3,3,1] => 4 [5,3,2,2] => 4 [5,3,2,1,1] => 4 [5,3,1,1,1,1] => 4 [5,2,2,2,1] => 4 [5,2,2,1,1,1] => 4 [5,2,1,1,1,1,1] => 4 [5,1,1,1,1,1,1,1] => 4 [4,4,4] => 3 [4,4,3,1] => 2 [4,4,2,2] => 3 [4,4,2,1,1] => 3 [4,4,1,1,1,1] => 3 [4,3,3,2] => 1 [4,3,3,1,1] => 2 [4,3,2,2,1] => 0 [4,3,2,1,1,1] => 0 [4,3,1,1,1,1,1] => 2 [4,2,2,2,2] => 3 [4,2,2,2,1,1] => 3 [4,2,2,1,1,1,1] => 3 [4,2,1,1,1,1,1,1] => 3 [4,1,1,1,1,1,1,1,1] => 3 [3,3,3,3] => 2 [3,3,3,2,1] => 0 [3,3,3,1,1,1] => 2 [3,3,2,2,2] => 1 [3,3,2,2,1,1] => 0 [3,3,2,1,1,1,1] => 0 [3,3,1,1,1,1,1,1] => 2 [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] => 2 [2,2,2,2,2,2] => 1 [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] => 12 [12,1] => 11 [11,2] => 10 [11,1,1] => 10 [10,3] => 9 [10,2,1] => 9 [10,1,1,1] => 9 [9,4] => 8 [9,3,1] => 8 [9,2,2] => 8 [9,2,1,1] => 8 [9,1,1,1,1] => 8 [8,5] => 7 [8,4,1] => 7 [8,3,2] => 7 [8,3,1,1] => 7 [8,2,2,1] => 7 [8,2,1,1,1] => 7 [8,1,1,1,1,1] => 7 [7,6] => 5 [7,5,1] => 6 [7,4,2] => 6 [7,4,1,1] => 6 [7,3,3] => 6 [7,3,2,1] => 6 [7,3,1,1,1] => 6 [7,2,2,2] => 6 [7,2,2,1,1] => 6 [7,2,1,1,1,1] => 6 [7,1,1,1,1,1,1] => 6 [6,6,1] => 5 [6,5,2] => 4 [6,5,1,1] => 4 [6,4,3] => 5 [6,4,2,1] => 5 [6,4,1,1,1] => 5 [6,3,3,1] => 5 [6,3,2,2] => 5 [6,3,2,1,1] => 5 [6,3,1,1,1,1] => 5 [6,2,2,2,1] => 5 [6,2,2,1,1,1] => 5 [6,2,1,1,1,1,1] => 5 [6,1,1,1,1,1,1,1] => 5 [5,5,3] => 4 [5,5,2,1] => 4 [5,5,1,1,1] => 4 [5,4,4] => 3 [5,4,3,1] => 2 [5,4,2,2] => 3 [5,4,2,1,1] => 3 [5,4,1,1,1,1] => 3 [5,3,3,2] => 4 [5,3,3,1,1] => 4 [5,3,2,2,1] => 4 [5,3,2,1,1,1] => 4 [5,3,1,1,1,1,1] => 4 [5,2,2,2,2] => 4 [5,2,2,2,1,1] => 4 [5,2,2,1,1,1,1] => 4 [5,2,1,1,1,1,1,1] => 4 [5,1,1,1,1,1,1,1,1] => 4 [4,4,4,1] => 3 [4,4,3,2] => 1 [4,4,3,1,1] => 2 [4,4,2,2,1] => 3 [4,4,2,1,1,1] => 3 [4,4,1,1,1,1,1] => 3 [4,3,3,3] => 2 [4,3,3,2,1] => 0 [4,3,3,1,1,1] => 2 [4,3,2,2,2] => 1 [4,3,2,2,1,1] => 0 [4,3,2,1,1,1,1] => 0 [4,3,1,1,1,1,1,1] => 2 [4,2,2,2,2,1] => 3 [4,2,2,2,1,1,1] => 3 [4,2,2,1,1,1,1,1] => 3 [4,2,1,1,1,1,1,1,1] => 3 [4,1,1,1,1,1,1,1,1,1] => 3 [3,3,3,3,1] => 2 [3,3,3,2,2] => 1 [3,3,3,2,1,1] => 0 [3,3,3,1,1,1,1] => 2 [3,3,2,2,2,1] => 0 [3,3,2,2,1,1,1] => 0 [3,3,2,1,1,1,1,1] => 0 [3,3,1,1,1,1,1,1,1] => 2 [3,2,2,2,2,2] => 1 [3,2,2,2,2,1,1] => 0 [3,2,2,2,1,1,1,1] => 0 [3,2,2,1,1,1,1,1,1] => 0 [3,2,1,1,1,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1,1,1,1] => 2 [2,2,2,2,2,2,1] => 0 [2,2,2,2,2,1,1,1] => 0 [2,2,2,2,1,1,1,1,1] => 0 [2,2,2,1,1,1,1,1,1,1] => 0 [2,2,1,1,1,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1,1,1,1] => 0 [14] => 13 [13,1] => 12 [12,2] => 11 [12,1,1] => 11 [11,3] => 10 [11,2,1] => 10 [11,1,1,1] => 10 [10,4] => 9 [10,3,1] => 9 [10,2,2] => 9 [10,2,1,1] => 9 [10,1,1,1,1] => 9 [9,5] => 8 [9,4,1] => 8 [9,3,2] => 8 [9,3,1,1] => 8 [9,2,2,1] => 8 [9,2,1,1,1] => 8 [9,1,1,1,1,1] => 8 [8,6] => 7 [8,5,1] => 7 [8,4,2] => 7 [8,4,1,1] => 7 [8,3,3] => 7 [8,3,2,1] => 7 [8,3,1,1,1] => 7 [8,2,2,2] => 7 [8,2,2,1,1] => 7 [8,2,1,1,1,1] => 7 [8,1,1,1,1,1,1] => 7 [7,7] => 6 [7,6,1] => 5 [7,5,2] => 6 [7,5,1,1] => 6 [7,4,3] => 6 [7,4,2,1] => 6 [7,4,1,1,1] => 6 [7,3,3,1] => 6 [7,3,2,2] => 6 [7,3,2,1,1] => 6 [7,3,1,1,1,1] => 6 [7,2,2,2,1] => 6 [7,2,2,1,1,1] => 6 [7,2,1,1,1,1,1] => 6 [7,1,1,1,1,1,1,1] => 6 [6,6,2] => 5 [6,6,1,1] => 5 [6,5,3] => 4 [6,5,2,1] => 4 [6,5,1,1,1] => 4 [6,4,4] => 5 [6,4,3,1] => 5 [6,4,2,2] => 5 [6,4,2,1,1] => 5 [6,4,1,1,1,1] => 5 [6,3,3,2] => 5 [6,3,3,1,1] => 5 [6,3,2,2,1] => 5 [6,3,2,1,1,1] => 5 [6,3,1,1,1,1,1] => 5 [6,2,2,2,2] => 5 [6,2,2,2,1,1] => 5 [6,2,2,1,1,1,1] => 5 [6,2,1,1,1,1,1,1] => 5 [6,1,1,1,1,1,1,1,1] => 5 [5,5,4] => 3 [5,5,3,1] => 4 [5,5,2,2] => 4 [5,5,2,1,1] => 4 [5,5,1,1,1,1] => 4 [5,4,4,1] => 3 [5,4,3,2] => 1 [5,4,3,1,1] => 2 [5,4,2,2,1] => 3 [5,4,2,1,1,1] => 3 [5,4,1,1,1,1,1] => 3 [5,3,3,3] => 4 [5,3,3,2,1] => 4 [5,3,3,1,1,1] => 4 [5,3,2,2,2] => 4 [5,3,2,2,1,1] => 4 [5,3,2,1,1,1,1] => 4 [5,3,1,1,1,1,1,1] => 4 [5,2,2,2,2,1] => 4 [5,2,2,2,1,1,1] => 4 [5,2,2,1,1,1,1,1] => 4 [5,2,1,1,1,1,1,1,1] => 4 [5,1,1,1,1,1,1,1,1,1] => 4 [4,4,4,2] => 3 [4,4,4,1,1] => 3 [4,4,3,3] => 2 [4,4,3,2,1] => 0 [4,4,3,1,1,1] => 2 [4,4,2,2,2] => 3 [4,4,2,2,1,1] => 3 [4,4,2,1,1,1,1] => 3 [4,4,1,1,1,1,1,1] => 3 [4,3,3,3,1] => 2 [4,3,3,2,2] => 1 [4,3,3,2,1,1] => 0 [4,3,3,1,1,1,1] => 2 [4,3,2,2,2,1] => 0 [4,3,2,2,1,1,1] => 0 [4,3,2,1,1,1,1,1] => 0 [4,3,1,1,1,1,1,1,1] => 2 [4,2,2,2,2,2] => 3 [4,2,2,2,2,1,1] => 3 [4,2,2,2,1,1,1,1] => 3 [4,2,2,1,1,1,1,1,1] => 3 [4,2,1,1,1,1,1,1,1,1] => 3 [4,1,1,1,1,1,1,1,1,1,1] => 3 [3,3,3,3,2] => 1 [3,3,3,3,1,1] => 2 [3,3,3,2,2,1] => 0 [3,3,3,2,1,1,1] => 0 [3,3,3,1,1,1,1,1] => 2 [3,3,2,2,2,2] => 1 [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] => 2 [3,2,2,2,2,2,1] => 0 [3,2,2,2,2,1,1,1] => 0 [3,2,2,2,1,1,1,1,1] => 0 [3,2,2,1,1,1,1,1,1,1] => 0 [3,2,1,1,1,1,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1,1,1,1,1] => 2 [2,2,2,2,2,2,2] => 1 [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] => 14 [14,1] => 13 [13,2] => 12 [13,1,1] => 12 [12,3] => 11 [12,2,1] => 11 [12,1,1,1] => 11 [11,4] => 10 [11,3,1] => 10 [11,2,2] => 10 [11,2,1,1] => 10 [11,1,1,1,1] => 10 [10,5] => 9 [10,4,1] => 9 [10,3,2] => 9 [10,3,1,1] => 9 [10,2,2,1] => 9 [10,2,1,1,1] => 9 [10,1,1,1,1,1] => 9 [9,6] => 8 [9,5,1] => 8 [9,4,2] => 8 [9,4,1,1] => 8 [9,3,3] => 8 [9,3,2,1] => 8 [9,3,1,1,1] => 8 [9,2,2,2] => 8 [9,2,2,1,1] => 8 [9,2,1,1,1,1] => 8 [9,1,1,1,1,1,1] => 8 [8,7] => 6 [8,6,1] => 7 [8,5,2] => 7 [8,5,1,1] => 7 [8,4,3] => 7 [8,4,2,1] => 7 [8,4,1,1,1] => 7 [8,3,3,1] => 7 [8,3,2,2] => 7 [8,3,2,1,1] => 7 [8,3,1,1,1,1] => 7 [8,2,2,2,1] => 7 [8,2,2,1,1,1] => 7 [8,2,1,1,1,1,1] => 7 [8,1,1,1,1,1,1,1] => 7 [7,7,1] => 6 [7,6,2] => 5 [7,6,1,1] => 5 [7,5,3] => 6 [7,5,2,1] => 6 [7,5,1,1,1] => 6 [7,4,4] => 6 [7,4,3,1] => 6 [7,4,2,2] => 6 [7,4,2,1,1] => 6 [7,4,1,1,1,1] => 6 [7,3,3,2] => 6 [7,3,3,1,1] => 6 [7,3,2,2,1] => 6 [7,3,2,1,1,1] => 6 [7,3,1,1,1,1,1] => 6 [7,2,2,2,2] => 6 [7,2,2,2,1,1] => 6 [7,2,2,1,1,1,1] => 6 [7,2,1,1,1,1,1,1] => 6 [7,1,1,1,1,1,1,1,1] => 6 [6,6,3] => 5 [6,6,2,1] => 5 [6,6,1,1,1] => 5 [6,5,4] => 3 [6,5,3,1] => 4 [6,5,2,2] => 4 [6,5,2,1,1] => 4 [6,5,1,1,1,1] => 4 [6,4,4,1] => 5 [6,4,3,2] => 5 [6,4,3,1,1] => 5 [6,4,2,2,1] => 5 [6,4,2,1,1,1] => 5 [6,4,1,1,1,1,1] => 5 [6,3,3,3] => 5 [6,3,3,2,1] => 5 [6,3,3,1,1,1] => 5 [6,3,2,2,2] => 5 [6,3,2,2,1,1] => 5 [6,3,2,1,1,1,1] => 5 [6,3,1,1,1,1,1,1] => 5 [6,2,2,2,2,1] => 5 [6,2,2,2,1,1,1] => 5 [6,2,2,1,1,1,1,1] => 5 [6,2,1,1,1,1,1,1,1] => 5 [6,1,1,1,1,1,1,1,1,1] => 5 [5,5,5] => 4 [5,5,4,1] => 3 [5,5,3,2] => 4 [5,5,3,1,1] => 4 [5,5,2,2,1] => 4 [5,5,2,1,1,1] => 4 [5,5,1,1,1,1,1] => 4 [5,4,4,2] => 3 [5,4,4,1,1] => 3 [5,4,3,3] => 2 [5,4,3,2,1] => 0 [5,4,3,1,1,1] => 2 [5,4,2,2,2] => 3 [5,4,2,2,1,1] => 3 [5,4,2,1,1,1,1] => 3 [5,4,1,1,1,1,1,1] => 3 [5,3,3,3,1] => 4 [5,3,3,2,2] => 4 [5,3,3,2,1,1] => 4 [5,3,3,1,1,1,1] => 4 [5,3,2,2,2,1] => 4 [5,3,2,2,1,1,1] => 4 [5,3,2,1,1,1,1,1] => 4 [5,3,1,1,1,1,1,1,1] => 4 [5,2,2,2,2,2] => 4 [5,2,2,2,2,1,1] => 4 [5,2,2,2,1,1,1,1] => 4 [5,2,2,1,1,1,1,1,1] => 4 [5,2,1,1,1,1,1,1,1,1] => 4 [5,1,1,1,1,1,1,1,1,1,1] => 4 [4,4,4,3] => 2 [4,4,4,2,1] => 3 [4,4,4,1,1,1] => 3 [4,4,3,3,1] => 2 [4,4,3,2,2] => 1 [4,4,3,2,1,1] => 0 [4,4,3,1,1,1,1] => 2 [4,4,2,2,2,1] => 3 [4,4,2,2,1,1,1] => 3 [4,4,2,1,1,1,1,1] => 3 [4,4,1,1,1,1,1,1,1] => 3 [4,3,3,3,2] => 1 [4,3,3,3,1,1] => 2 [4,3,3,2,2,1] => 0 [4,3,3,2,1,1,1] => 0 [4,3,3,1,1,1,1,1] => 2 [4,3,2,2,2,2] => 1 [4,3,2,2,2,1,1] => 0 [4,3,2,2,1,1,1,1] => 0 [4,3,2,1,1,1,1,1,1] => 0 [4,3,1,1,1,1,1,1,1,1] => 2 [4,2,2,2,2,2,1] => 3 [4,2,2,2,2,1,1,1] => 3 [4,2,2,2,1,1,1,1,1] => 3 [4,2,2,1,1,1,1,1,1,1] => 3 [4,2,1,1,1,1,1,1,1,1,1] => 3 [4,1,1,1,1,1,1,1,1,1,1,1] => 3 [3,3,3,3,3] => 2 [3,3,3,3,2,1] => 0 [3,3,3,3,1,1,1] => 2 [3,3,3,2,2,2] => 1 [3,3,3,2,2,1,1] => 0 [3,3,3,2,1,1,1,1] => 0 [3,3,3,1,1,1,1,1,1] => 2 [3,3,2,2,2,2,1] => 0 [3,3,2,2,2,1,1,1] => 0 [3,3,2,2,1,1,1,1,1] => 0 [3,3,2,1,1,1,1,1,1,1] => 0 [3,3,1,1,1,1,1,1,1,1,1] => 2 [3,2,2,2,2,2,2] => 1 [3,2,2,2,2,2,1,1] => 0 [3,2,2,2,2,1,1,1,1] => 0 [3,2,2,2,1,1,1,1,1,1] => 0 [3,2,2,1,1,1,1,1,1,1,1] => 0 [3,2,1,1,1,1,1,1,1,1,1,1] => 0 [3,1,1,1,1,1,1,1,1,1,1,1,1] => 2 [2,2,2,2,2,2,2,1] => 0 [2,2,2,2,2,2,1,1,1] => 0 [2,2,2,2,2,1,1,1,1,1] => 0 [2,2,2,2,1,1,1,1,1,1,1] => 0 [2,2,2,1,1,1,1,1,1,1,1,1] => 0 [2,2,1,1,1,1,1,1,1,1,1,1,1] => 0 [2,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1] => 0 [16] => 15 [15,1] => 14 [14,2] => 13 [14,1,1] => 13 [13,3] => 12 [13,2,1] => 12 [13,1,1,1] => 12 [12,4] => 11 [12,3,1] => 11 [12,2,2] => 11 [12,2,1,1] => 11 [12,1,1,1,1] => 11 [11,5] => 10 [11,4,1] => 10 [11,3,2] => 10 [11,3,1,1] => 10 [11,2,2,1] => 10 [11,2,1,1,1] => 10 [11,1,1,1,1,1] => 10 [10,6] => 9 [10,5,1] => 9 [10,4,2] => 9 [10,4,1,1] => 9 [10,3,3] => 9 [10,3,2,1] => 9 [10,3,1,1,1] => 9 [10,2,2,2] => 9 [10,2,2,1,1] => 9 [10,2,1,1,1,1] => 9 [10,1,1,1,1,1,1] => 9 [9,7] => 8 [9,6,1] => 8 [9,5,2] => 8 [9,5,1,1] => 8 [9,4,3] => 8 [9,4,2,1] => 8 [9,4,1,1,1] => 8 [9,3,3,1] => 8 [9,3,2,2] => 8 [9,3,2,1,1] => 8 [9,3,1,1,1,1] => 8 [9,2,2,2,1] => 8 [9,2,2,1,1,1] => 8 [9,2,1,1,1,1,1] => 8 [9,1,1,1,1,1,1,1] => 8 [8,8] => 7 [8,7,1] => 6 [8,6,2] => 7 [8,6,1,1] => 7 [8,5,3] => 7 [8,5,2,1] => 7 [8,5,1,1,1] => 7 [8,4,4] => 7 [8,4,3,1] => 7 [8,4,2,2] => 7 [8,4,2,1,1] => 7 [8,4,1,1,1,1] => 7 [8,3,3,2] => 7 [8,3,3,1,1] => 7 [8,3,2,2,1] => 7 [8,3,2,1,1,1] => 7 [8,3,1,1,1,1,1] => 7 [8,2,2,2,2] => 7 [8,2,2,2,1,1] => 7 [8,2,2,1,1,1,1] => 7 [8,2,1,1,1,1,1,1] => 7 [8,1,1,1,1,1,1,1,1] => 7 [7,7,2] => 6 [7,7,1,1] => 6 [7,6,3] => 5 [7,6,2,1] => 5 [7,6,1,1,1] => 5 [7,5,4] => 6 [7,5,3,1] => 6 [7,5,2,2] => 6 [7,5,2,1,1] => 6 [7,5,1,1,1,1] => 6 [7,4,4,1] => 6 [7,4,3,2] => 6 [7,4,3,1,1] => 6 [7,4,2,2,1] => 6 [7,4,2,1,1,1] => 6 [7,4,1,1,1,1,1] => 6 [7,3,3,3] => 6 [7,3,3,2,1] => 6 [7,3,3,1,1,1] => 6 [7,3,2,2,2] => 6 [7,3,2,2,1,1] => 6 [7,3,2,1,1,1,1] => 6 [7,3,1,1,1,1,1,1] => 6 [7,2,2,2,2,1] => 6 [7,2,2,2,1,1,1] => 6 [7,2,2,1,1,1,1,1] => 6 [7,2,1,1,1,1,1,1,1] => 6 [7,1,1,1,1,1,1,1,1,1] => 6 [6,6,4] => 5 [6,6,3,1] => 5 [6,6,2,2] => 5 [6,6,2,1,1] => 5 [6,6,1,1,1,1] => 5 [6,5,5] => 4 [6,5,4,1] => 3 [6,5,3,2] => 4 [6,5,3,1,1] => 4 [6,5,2,2,1] => 4 [6,5,2,1,1,1] => 4 [6,5,1,1,1,1,1] => 4 [6,4,4,2] => 5 [6,4,4,1,1] => 5 [6,4,3,3] => 5 [6,4,3,2,1] => 5 [6,4,3,1,1,1] => 5 [6,4,2,2,2] => 5 [6,4,2,2,1,1] => 5 [6,4,2,1,1,1,1] => 5 [6,4,1,1,1,1,1,1] => 5 [6,3,3,3,1] => 5 [6,3,3,2,2] => 5 [6,3,3,2,1,1] => 5 [6,3,3,1,1,1,1] => 5 [6,3,2,2,2,1] => 5 [6,3,2,2,1,1,1] => 5 [6,3,2,1,1,1,1,1] => 5 [6,3,1,1,1,1,1,1,1] => 5 [6,2,2,2,2,2] => 5 [6,2,2,2,2,1,1] => 5 [6,2,2,2,1,1,1,1] => 5 [6,2,2,1,1,1,1,1,1] => 5 [6,2,1,1,1,1,1,1,1,1] => 5 [6,1,1,1,1,1,1,1,1,1,1] => 5 [5,5,5,1] => 4 [5,5,4,2] => 3 [5,5,4,1,1] => 3 [5,5,3,3] => 4 [5,5,3,2,1] => 4 [5,5,3,1,1,1] => 4 [5,5,2,2,2] => 4 [5,5,2,2,1,1] => 4 [5,5,2,1,1,1,1] => 4 [5,5,1,1,1,1,1,1] => 4 [5,4,4,3] => 2 [5,4,4,2,1] => 3 [5,4,4,1,1,1] => 3 [5,4,3,3,1] => 2 [5,4,3,2,2] => 1 [5,4,3,2,1,1] => 0 [5,4,3,1,1,1,1] => 2 [5,4,2,2,2,1] => 3 [5,4,2,2,1,1,1] => 3 [5,4,2,1,1,1,1,1] => 3 [5,4,1,1,1,1,1,1,1] => 3 [5,3,3,3,2] => 4 [5,3,3,3,1,1] => 4 [5,3,3,2,2,1] => 4 [5,3,3,2,1,1,1] => 4 [5,3,3,1,1,1,1,1] => 4 [5,3,2,2,2,2] => 4 [5,3,2,2,2,1,1] => 4 [5,3,2,2,1,1,1,1] => 4 [5,3,2,1,1,1,1,1,1] => 4 [5,3,1,1,1,1,1,1,1,1] => 4 [5,2,2,2,2,2,1] => 4 [5,2,2,2,2,1,1,1] => 4 [5,2,2,2,1,1,1,1,1] => 4 [5,2,2,1,1,1,1,1,1,1] => 4 [5,2,1,1,1,1,1,1,1,1,1] => 4 [5,1,1,1,1,1,1,1,1,1,1,1] => 4 [4,4,4,4] => 3 [4,4,4,3,1] => 2 [4,4,4,2,2] => 3 [4,4,4,2,1,1] => 3 [4,4,4,1,1,1,1] => 3 [4,4,3,3,2] => 1 [4,4,3,3,1,1] => 2 [4,4,3,2,2,1] => 0 [4,4,3,2,1,1,1] => 0 [4,4,3,1,1,1,1,1] => 2 [4,4,2,2,2,2] => 3 [4,4,2,2,2,1,1] => 3 [4,4,2,2,1,1,1,1] => 3 [4,4,2,1,1,1,1,1,1] => 3 [4,4,1,1,1,1,1,1,1,1] => 3 [4,3,3,3,3] => 2 [4,3,3,3,2,1] => 0 [4,3,3,3,1,1,1] => 2 [4,3,3,2,2,2] => 1 [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] => 2 [4,3,2,2,2,2,1] => 0 [4,3,2,2,2,1,1,1] => 0 [4,3,2,2,1,1,1,1,1] => 0 [4,3,2,1,1,1,1,1,1,1] => 0 [4,3,1,1,1,1,1,1,1,1,1] => 2 [4,2,2,2,2,2,2] => 3 [4,2,2,2,2,2,1,1] => 3 [4,2,2,2,2,1,1,1,1] => 3 [4,2,2,2,1,1,1,1,1,1] => 3 [4,2,2,1,1,1,1,1,1,1,1] => 3 [4,2,1,1,1,1,1,1,1,1,1,1] => 3 [4,1,1,1,1,1,1,1,1,1,1,1,1] => 3 [3,3,3,3,3,1] => 2 [3,3,3,3,2,2] => 1 [3,3,3,3,2,1,1] => 0 [3,3,3,3,1,1,1,1] => 2 [3,3,3,2,2,2,1] => 0 [3,3,3,2,2,1,1,1] => 0 [3,3,3,2,1,1,1,1,1] => 0 [3,3,3,1,1,1,1,1,1,1] => 2 [3,3,2,2,2,2,2] => 1 [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] => 2 [3,2,2,2,2,2,2,1] => 0 [3,2,2,2,2,2,1,1,1] => 0 [3,2,2,2,2,1,1,1,1,1] => 0 [3,2,2,2,1,1,1,1,1,1,1] => 0 [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] => 0 [3,1,1,1,1,1,1,1,1,1,1,1,1,1] => 2 [2,2,2,2,2,2,2,2] => 1 [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] => 16 [16,1] => 15 [15,2] => 14 [15,1,1] => 14 [14,3] => 13 [14,2,1] => 13 [14,1,1,1] => 13 [13,4] => 12 [13,3,1] => 12 [13,2,2] => 12 [13,2,1,1] => 12 [13,1,1,1,1] => 12 [12,5] => 11 [12,4,1] => 11 [12,3,2] => 11 [12,3,1,1] => 11 [12,2,2,1] => 11 [12,2,1,1,1] => 11 [12,1,1,1,1,1] => 11 [11,6] => 10 [11,5,1] => 10 [11,4,2] => 10 [11,4,1,1] => 10 [11,3,3] => 10 [11,3,2,1] => 10 [11,3,1,1,1] => 10 [11,2,2,2] => 10 [11,2,2,1,1] => 10 [11,2,1,1,1,1] => 10 [11,1,1,1,1,1,1] => 10 [10,7] => 9 [10,6,1] => 9 [10,5,2] => 9 [10,5,1,1] => 9 [10,4,3] => 9 [10,4,2,1] => 9 [10,4,1,1,1] => 9 [10,3,3,1] => 9 [10,3,2,2] => 9 [10,3,2,1,1] => 9 [10,3,1,1,1,1] => 9 [10,2,2,2,1] => 9 [10,2,2,1,1,1] => 9 [10,2,1,1,1,1,1] => 9 [10,1,1,1,1,1,1,1] => 9 [9,8] => 7 [9,7,1] => 8 [9,6,2] => 8 [9,6,1,1] => 8 [9,5,3] => 8 [9,5,2,1] => 8 [9,5,1,1,1] => 8 [9,4,4] => 8 [9,4,3,1] => 8 [9,4,2,2] => 8 [9,4,2,1,1] => 8 [9,4,1,1,1,1] => 8 [9,3,3,2] => 8 [9,3,3,1,1] => 8 [9,3,2,2,1] => 8 [9,3,2,1,1,1] => 8 [9,3,1,1,1,1,1] => 8 [9,2,2,2,2] => 8 [9,2,2,2,1,1] => 8 [9,2,2,1,1,1,1] => 8 [9,2,1,1,1,1,1,1] => 8 [9,1,1,1,1,1,1,1,1] => 8 [8,8,1] => 7 [8,7,2] => 6 [8,7,1,1] => 6 [8,6,3] => 7 [8,6,2,1] => 7 [8,6,1,1,1] => 7 [8,5,4] => 7 [8,5,3,1] => 7 [8,5,2,2] => 7 [8,5,2,1,1] => 7 [8,5,1,1,1,1] => 7 [8,4,4,1] => 7 [8,4,3,2] => 7 [8,4,3,1,1] => 7 [8,4,2,2,1] => 7 [8,4,2,1,1,1] => 7 [8,4,1,1,1,1,1] => 7 [8,3,3,3] => 7 [8,3,3,2,1] => 7 [8,3,3,1,1,1] => 7 [8,3,2,2,2] => 7 [8,3,2,2,1,1] => 7 [8,3,2,1,1,1,1] => 7 [8,3,1,1,1,1,1,1] => 7 [8,2,2,2,2,1] => 7 [8,2,2,2,1,1,1] => 7 [8,2,2,1,1,1,1,1] => 7 [8,2,1,1,1,1,1,1,1] => 7 [8,1,1,1,1,1,1,1,1,1] => 7 [7,7,3] => 6 [7,7,2,1] => 6 [7,7,1,1,1] => 6 [7,6,4] => 5 [7,6,3,1] => 5 [7,6,2,2] => 5 [7,6,2,1,1] => 5 [7,6,1,1,1,1] => 5 [7,5,5] => 6 [7,5,4,1] => 6 [7,5,3,2] => 6 [7,5,3,1,1] => 6 [7,5,2,2,1] => 6 [7,5,2,1,1,1] => 6 [7,5,1,1,1,1,1] => 6 [7,4,4,2] => 6 [7,4,4,1,1] => 6 [7,4,3,3] => 6 [7,4,3,2,1] => 6 [7,4,3,1,1,1] => 6 [7,4,2,2,2] => 6 [7,4,2,2,1,1] => 6 [7,4,2,1,1,1,1] => 6 [7,4,1,1,1,1,1,1] => 6 [7,3,3,3,1] => 6 [7,3,3,2,2] => 6 [7,3,3,2,1,1] => 6 [7,3,3,1,1,1,1] => 6 [7,3,2,2,2,1] => 6 [7,3,2,2,1,1,1] => 6 [7,3,2,1,1,1,1,1] => 6 [7,3,1,1,1,1,1,1,1] => 6 [7,2,2,2,2,2] => 6 [7,2,2,2,2,1,1] => 6 [7,2,2,2,1,1,1,1] => 6 [7,2,2,1,1,1,1,1,1] => 6 [7,2,1,1,1,1,1,1,1,1] => 6 [7,1,1,1,1,1,1,1,1,1,1] => 6 [6,6,5] => 4 [6,6,4,1] => 5 [6,6,3,2] => 5 [6,6,3,1,1] => 5 [6,6,2,2,1] => 5 [6,6,2,1,1,1] => 5 [6,6,1,1,1,1,1] => 5 [6,5,5,1] => 4 [6,5,4,2] => 3 [6,5,4,1,1] => 3 [6,5,3,3] => 4 [6,5,3,2,1] => 4 [6,5,3,1,1,1] => 4 [6,5,2,2,2] => 4 [6,5,2,2,1,1] => 4 [6,5,2,1,1,1,1] => 4 [6,5,1,1,1,1,1,1] => 4 [6,4,4,3] => 5 [6,4,4,2,1] => 5 [6,4,4,1,1,1] => 5 [6,4,3,3,1] => 5 [6,4,3,2,2] => 5 [6,4,3,2,1,1] => 5 [6,4,3,1,1,1,1] => 5 [6,4,2,2,2,1] => 5 [6,4,2,2,1,1,1] => 5 [6,4,2,1,1,1,1,1] => 5 [6,4,1,1,1,1,1,1,1] => 5 [6,3,3,3,2] => 5 [6,3,3,3,1,1] => 5 [6,3,3,2,2,1] => 5 [6,3,3,2,1,1,1] => 5 [6,3,3,1,1,1,1,1] => 5 [6,3,2,2,2,2] => 5 [6,3,2,2,2,1,1] => 5 [6,3,2,2,1,1,1,1] => 5 [6,3,2,1,1,1,1,1,1] => 5 [6,3,1,1,1,1,1,1,1,1] => 5 [6,2,2,2,2,2,1] => 5 [6,2,2,2,2,1,1,1] => 5 [6,2,2,2,1,1,1,1,1] => 5 [6,2,2,1,1,1,1,1,1,1] => 5 [6,2,1,1,1,1,1,1,1,1,1] => 5 [6,1,1,1,1,1,1,1,1,1,1,1] => 5 [5,5,5,2] => 4 [5,5,5,1,1] => 4 [5,5,4,3] => 2 [5,5,4,2,1] => 3 [5,5,4,1,1,1] => 3 [5,5,3,3,1] => 4 [5,5,3,2,2] => 4 [5,5,3,2,1,1] => 4 [5,5,3,1,1,1,1] => 4 [5,5,2,2,2,1] => 4 [5,5,2,2,1,1,1] => 4 [5,5,2,1,1,1,1,1] => 4 [5,5,1,1,1,1,1,1,1] => 4 [5,4,4,4] => 3 [5,4,4,3,1] => 2 [5,4,4,2,2] => 3 [5,4,4,2,1,1] => 3 [5,4,4,1,1,1,1] => 3 [5,4,3,3,2] => 1 [5,4,3,3,1,1] => 2 [5,4,3,2,2,1] => 0 [5,4,3,2,1,1,1] => 0 [5,4,3,1,1,1,1,1] => 2 [5,4,2,2,2,2] => 3 [5,4,2,2,2,1,1] => 3 [5,4,2,2,1,1,1,1] => 3 [5,4,2,1,1,1,1,1,1] => 3 [5,4,1,1,1,1,1,1,1,1] => 3 [5,3,3,3,3] => 4 [5,3,3,3,2,1] => 4 [5,3,3,3,1,1,1] => 4 [5,3,3,2,2,2] => 4 [5,3,3,2,2,1,1] => 4 [5,3,3,2,1,1,1,1] => 4 [5,3,3,1,1,1,1,1,1] => 4 [5,3,2,2,2,2,1] => 4 [5,3,2,2,2,1,1,1] => 4 [5,3,2,2,1,1,1,1,1] => 4 [5,3,2,1,1,1,1,1,1,1] => 4 [5,3,1,1,1,1,1,1,1,1,1] => 4 [5,2,2,2,2,2,2] => 4 [5,2,2,2,2,2,1,1] => 4 [5,2,2,2,2,1,1,1,1] => 4 [5,2,2,2,1,1,1,1,1,1] => 4 [5,2,2,1,1,1,1,1,1,1,1] => 4 [5,2,1,1,1,1,1,1,1,1,1,1] => 4 [5,1,1,1,1,1,1,1,1,1,1,1,1] => 4 [4,4,4,4,1] => 3 [4,4,4,3,2] => 1 [4,4,4,3,1,1] => 2 [4,4,4,2,2,1] => 3 [4,4,4,2,1,1,1] => 3 [4,4,4,1,1,1,1,1] => 3 [4,4,3,3,3] => 2 [4,4,3,3,2,1] => 0 [4,4,3,3,1,1,1] => 2 [4,4,3,2,2,2] => 1 [4,4,3,2,2,1,1] => 0 [4,4,3,2,1,1,1,1] => 0 [4,4,3,1,1,1,1,1,1] => 2 [4,4,2,2,2,2,1] => 3 [4,4,2,2,2,1,1,1] => 3 [4,4,2,2,1,1,1,1,1] => 3 [4,4,2,1,1,1,1,1,1,1] => 3 [4,4,1,1,1,1,1,1,1,1,1] => 3 [4,3,3,3,3,1] => 2 [4,3,3,3,2,2] => 1 [4,3,3,3,2,1,1] => 0 [4,3,3,3,1,1,1,1] => 2 [4,3,3,2,2,2,1] => 0 [4,3,3,2,2,1,1,1] => 0 [4,3,3,2,1,1,1,1,1] => 0 [4,3,3,1,1,1,1,1,1,1] => 2 [4,3,2,2,2,2,2] => 1 [4,3,2,2,2,2,1,1] => 0 [4,3,2,2,2,1,1,1,1] => 0 [4,3,2,2,1,1,1,1,1,1] => 0 [4,3,2,1,1,1,1,1,1,1,1] => 0 [4,3,1,1,1,1,1,1,1,1,1,1] => 2 [4,2,2,2,2,2,2,1] => 3 [4,2,2,2,2,2,1,1,1] => 3 [4,2,2,2,2,1,1,1,1,1] => 3 [4,2,2,2,1,1,1,1,1,1,1] => 3 [4,2,2,1,1,1,1,1,1,1,1,1] => 3 [4,2,1,1,1,1,1,1,1,1,1,1,1] => 3 [4,1,1,1,1,1,1,1,1,1,1,1,1,1] => 3 [3,3,3,3,3,2] => 1 [3,3,3,3,3,1,1] => 2 [3,3,3,3,2,2,1] => 0 [3,3,3,3,2,1,1,1] => 0 [3,3,3,3,1,1,1,1,1] => 2 [3,3,3,2,2,2,2] => 1 [3,3,3,2,2,2,1,1] => 0 [3,3,3,2,2,1,1,1,1] => 0 [3,3,3,2,1,1,1,1,1,1] => 0 [3,3,3,1,1,1,1,1,1,1,1] => 2 [3,3,2,2,2,2,2,1] => 0 [3,3,2,2,2,2,1,1,1] => 0 [3,3,2,2,2,1,1,1,1,1] => 0 [3,3,2,2,1,1,1,1,1,1,1] => 0 [3,3,2,1,1,1,1,1,1,1,1,1] => 0 [3,3,1,1,1,1,1,1,1,1,1,1,1] => 2 [3,2,2,2,2,2,2,2] => 1 [3,2,2,2,2,2,2,1,1] => 0 [3,2,2,2,2,2,1,1,1,1] => 0 [3,2,2,2,2,1,1,1,1,1,1] => 0 [3,2,2,2,1,1,1,1,1,1,1,1] => 0 [3,2,2,1,1,1,1,1,1,1,1,1,1] => 0 ----------------------------------------------------------------------------- Created: May 16, 2019 at 08:11 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Mar 05, 2020 at 14:14 by Martin Rubey