***************************************************************************** * 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: St000784 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The maximum of the length and the largest part of the integer partition. This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1]. ----------------------------------------------------------------------------- References: [1] Chow, T. Coloring a Ferrers diagram [[MathOverflow:203962]] ----------------------------------------------------------------------------- Code: def statistic(pi): if pi: return max(pi[0], len(pi)) return 0 ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => 1 [2] => 2 [1,1] => 2 [3] => 3 [2,1] => 2 [1,1,1] => 3 [4] => 4 [3,1] => 3 [2,2] => 2 [2,1,1] => 3 [1,1,1,1] => 4 [5] => 5 [4,1] => 4 [3,2] => 3 [3,1,1] => 3 [2,2,1] => 3 [2,1,1,1] => 4 [1,1,1,1,1] => 5 [6] => 6 [5,1] => 5 [4,2] => 4 [4,1,1] => 4 [3,3] => 3 [3,2,1] => 3 [3,1,1,1] => 4 [2,2,2] => 3 [2,2,1,1] => 4 [2,1,1,1,1] => 5 [1,1,1,1,1,1] => 6 [7] => 7 [6,1] => 6 [5,2] => 5 [5,1,1] => 5 [4,3] => 4 [4,2,1] => 4 [4,1,1,1] => 4 [3,3,1] => 3 [3,2,2] => 3 [3,2,1,1] => 4 [3,1,1,1,1] => 5 [2,2,2,1] => 4 [2,2,1,1,1] => 5 [2,1,1,1,1,1] => 6 [1,1,1,1,1,1,1] => 7 [8] => 8 [7,1] => 7 [6,2] => 6 [6,1,1] => 6 [5,3] => 5 [5,2,1] => 5 [5,1,1,1] => 5 [4,4] => 4 [4,3,1] => 4 [4,2,2] => 4 [4,2,1,1] => 4 [4,1,1,1,1] => 5 [3,3,2] => 3 [3,3,1,1] => 4 [3,2,2,1] => 4 [3,2,1,1,1] => 5 [3,1,1,1,1,1] => 6 [2,2,2,2] => 4 [2,2,2,1,1] => 5 [2,2,1,1,1,1] => 6 [2,1,1,1,1,1,1] => 7 [1,1,1,1,1,1,1,1] => 8 [9] => 9 [8,1] => 8 [7,2] => 7 [7,1,1] => 7 [6,3] => 6 [6,2,1] => 6 [6,1,1,1] => 6 [5,4] => 5 [5,3,1] => 5 [5,2,2] => 5 [5,2,1,1] => 5 [5,1,1,1,1] => 5 [4,4,1] => 4 [4,3,2] => 4 [4,3,1,1] => 4 [4,2,2,1] => 4 [4,2,1,1,1] => 5 [4,1,1,1,1,1] => 6 [3,3,3] => 3 [3,3,2,1] => 4 [3,3,1,1,1] => 5 [3,2,2,2] => 4 [3,2,2,1,1] => 5 [3,2,1,1,1,1] => 6 [3,1,1,1,1,1,1] => 7 [2,2,2,2,1] => 5 [2,2,2,1,1,1] => 6 [2,2,1,1,1,1,1] => 7 [2,1,1,1,1,1,1,1] => 8 [1,1,1,1,1,1,1,1,1] => 9 [10] => 10 [9,1] => 9 [8,2] => 8 [8,1,1] => 8 [7,3] => 7 [7,2,1] => 7 [7,1,1,1] => 7 [6,4] => 6 [6,3,1] => 6 [6,2,2] => 6 [6,2,1,1] => 6 [6,1,1,1,1] => 6 [5,5] => 5 [5,4,1] => 5 [5,3,2] => 5 [5,3,1,1] => 5 [5,2,2,1] => 5 [5,2,1,1,1] => 5 [5,1,1,1,1,1] => 6 [4,4,2] => 4 [4,4,1,1] => 4 [4,3,3] => 4 [4,3,2,1] => 4 [4,3,1,1,1] => 5 [4,2,2,2] => 4 [4,2,2,1,1] => 5 [4,2,1,1,1,1] => 6 [4,1,1,1,1,1,1] => 7 [3,3,3,1] => 4 [3,3,2,2] => 4 [3,3,2,1,1] => 5 [3,3,1,1,1,1] => 6 [3,2,2,2,1] => 5 [3,2,2,1,1,1] => 6 [3,2,1,1,1,1,1] => 7 [3,1,1,1,1,1,1,1] => 8 [2,2,2,2,2] => 5 [2,2,2,2,1,1] => 6 [2,2,2,1,1,1,1] => 7 [2,2,1,1,1,1,1,1] => 8 [2,1,1,1,1,1,1,1,1] => 9 [1,1,1,1,1,1,1,1,1,1] => 10 [11] => 11 [10,1] => 10 [9,2] => 9 [9,1,1] => 9 [8,3] => 8 [8,2,1] => 8 [8,1,1,1] => 8 [7,4] => 7 [7,3,1] => 7 [7,2,2] => 7 [7,2,1,1] => 7 [7,1,1,1,1] => 7 [6,5] => 6 [6,4,1] => 6 [6,3,2] => 6 [6,3,1,1] => 6 [6,2,2,1] => 6 [6,2,1,1,1] => 6 [6,1,1,1,1,1] => 6 [5,5,1] => 5 [5,4,2] => 5 [5,4,1,1] => 5 [5,3,3] => 5 [5,3,2,1] => 5 [5,3,1,1,1] => 5 [5,2,2,2] => 5 [5,2,2,1,1] => 5 [5,2,1,1,1,1] => 6 [5,1,1,1,1,1,1] => 7 [4,4,3] => 4 [4,4,2,1] => 4 [4,4,1,1,1] => 5 [4,3,3,1] => 4 [4,3,2,2] => 4 [4,3,2,1,1] => 5 [4,3,1,1,1,1] => 6 [4,2,2,2,1] => 5 [4,2,2,1,1,1] => 6 [4,2,1,1,1,1,1] => 7 [4,1,1,1,1,1,1,1] => 8 [3,3,3,2] => 4 [3,3,3,1,1] => 5 [3,3,2,2,1] => 5 [3,3,2,1,1,1] => 6 [3,3,1,1,1,1,1] => 7 [3,2,2,2,2] => 5 [3,2,2,2,1,1] => 6 [3,2,2,1,1,1,1] => 7 [3,2,1,1,1,1,1,1] => 8 [3,1,1,1,1,1,1,1,1] => 9 [2,2,2,2,2,1] => 6 [2,2,2,2,1,1,1] => 7 [2,2,2,1,1,1,1,1] => 8 [2,2,1,1,1,1,1,1,1] => 9 [2,1,1,1,1,1,1,1,1,1] => 10 [1,1,1,1,1,1,1,1,1,1,1] => 11 [12] => 12 [11,1] => 11 [10,2] => 10 [10,1,1] => 10 [9,3] => 9 [9,2,1] => 9 [9,1,1,1] => 9 [8,4] => 8 [8,3,1] => 8 [8,2,2] => 8 [8,2,1,1] => 8 [8,1,1,1,1] => 8 [7,5] => 7 [7,4,1] => 7 [7,3,2] => 7 [7,3,1,1] => 7 [7,2,2,1] => 7 [7,2,1,1,1] => 7 [7,1,1,1,1,1] => 7 [6,6] => 6 [6,5,1] => 6 [6,4,2] => 6 [6,4,1,1] => 6 [6,3,3] => 6 [6,3,2,1] => 6 [6,3,1,1,1] => 6 [6,2,2,2] => 6 [6,2,2,1,1] => 6 [6,2,1,1,1,1] => 6 [6,1,1,1,1,1,1] => 7 [5,5,2] => 5 [5,5,1,1] => 5 [5,4,3] => 5 [5,4,2,1] => 5 [5,4,1,1,1] => 5 [5,3,3,1] => 5 [5,3,2,2] => 5 [5,3,2,1,1] => 5 [5,3,1,1,1,1] => 6 [5,2,2,2,1] => 5 [5,2,2,1,1,1] => 6 [5,2,1,1,1,1,1] => 7 [5,1,1,1,1,1,1,1] => 8 [4,4,4] => 4 [4,4,3,1] => 4 [4,4,2,2] => 4 [4,4,2,1,1] => 5 [4,4,1,1,1,1] => 6 [4,3,3,2] => 4 [4,3,3,1,1] => 5 [4,3,2,2,1] => 5 [4,3,2,1,1,1] => 6 [4,3,1,1,1,1,1] => 7 [4,2,2,2,2] => 5 [4,2,2,2,1,1] => 6 [4,2,2,1,1,1,1] => 7 [4,2,1,1,1,1,1,1] => 8 [4,1,1,1,1,1,1,1,1] => 9 [3,3,3,3] => 4 [3,3,3,2,1] => 5 [3,3,3,1,1,1] => 6 [3,3,2,2,2] => 5 [3,3,2,2,1,1] => 6 [3,3,2,1,1,1,1] => 7 [3,3,1,1,1,1,1,1] => 8 [3,2,2,2,2,1] => 6 [3,2,2,2,1,1,1] => 7 [3,2,2,1,1,1,1,1] => 8 [3,2,1,1,1,1,1,1,1] => 9 [3,1,1,1,1,1,1,1,1,1] => 10 [2,2,2,2,2,2] => 6 [2,2,2,2,2,1,1] => 7 [2,2,2,2,1,1,1,1] => 8 [2,2,2,1,1,1,1,1,1] => 9 [2,2,1,1,1,1,1,1,1,1] => 10 [2,1,1,1,1,1,1,1,1,1,1] => 11 [1,1,1,1,1,1,1,1,1,1,1,1] => 12 ----------------------------------------------------------------------------- Created: Apr 19, 2017 at 09:45 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Oct 29, 2017 at 21:29 by Martin Rubey