***************************************************************************** * 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: St001214 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The aft of an integer partition. The aft is the size of the partition minus the length of the first row or column, whichever is larger. See also [[St000784]]. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(L): if L: return sum(L)-max(L[0],len(L)) else: return 0 ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => 0 [2] => 0 [1,1] => 0 [3] => 0 [2,1] => 1 [1,1,1] => 0 [4] => 0 [3,1] => 1 [2,2] => 2 [2,1,1] => 1 [1,1,1,1] => 0 [5] => 0 [4,1] => 1 [3,2] => 2 [3,1,1] => 2 [2,2,1] => 2 [2,1,1,1] => 1 [1,1,1,1,1] => 0 [6] => 0 [5,1] => 1 [4,2] => 2 [4,1,1] => 2 [3,3] => 3 [3,2,1] => 3 [3,1,1,1] => 2 [2,2,2] => 3 [2,2,1,1] => 2 [2,1,1,1,1] => 1 [1,1,1,1,1,1] => 0 [7] => 0 [6,1] => 1 [5,2] => 2 [5,1,1] => 2 [4,3] => 3 [4,2,1] => 3 [4,1,1,1] => 3 [3,3,1] => 4 [3,2,2] => 4 [3,2,1,1] => 3 [3,1,1,1,1] => 2 [2,2,2,1] => 3 [2,2,1,1,1] => 2 [2,1,1,1,1,1] => 1 [1,1,1,1,1,1,1] => 0 [8] => 0 [7,1] => 1 [6,2] => 2 [6,1,1] => 2 [5,3] => 3 [5,2,1] => 3 [5,1,1,1] => 3 [4,4] => 4 [4,3,1] => 4 [4,2,2] => 4 [4,2,1,1] => 4 [4,1,1,1,1] => 3 [3,3,2] => 5 [3,3,1,1] => 4 [3,2,2,1] => 4 [3,2,1,1,1] => 3 [3,1,1,1,1,1] => 2 [2,2,2,2] => 4 [2,2,2,1,1] => 3 [2,2,1,1,1,1] => 2 [2,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1] => 0 [9] => 0 [8,1] => 1 [7,2] => 2 [7,1,1] => 2 [6,3] => 3 [6,2,1] => 3 [6,1,1,1] => 3 [5,4] => 4 [5,3,1] => 4 [5,2,2] => 4 [5,2,1,1] => 4 [5,1,1,1,1] => 4 [4,4,1] => 5 [4,3,2] => 5 [4,3,1,1] => 5 [4,2,2,1] => 5 [4,2,1,1,1] => 4 [4,1,1,1,1,1] => 3 [3,3,3] => 6 [3,3,2,1] => 5 [3,3,1,1,1] => 4 [3,2,2,2] => 5 [3,2,2,1,1] => 4 [3,2,1,1,1,1] => 3 [3,1,1,1,1,1,1] => 2 [2,2,2,2,1] => 4 [2,2,2,1,1,1] => 3 [2,2,1,1,1,1,1] => 2 [2,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1] => 0 [10] => 0 [9,1] => 1 [8,2] => 2 [8,1,1] => 2 [7,3] => 3 [7,2,1] => 3 [7,1,1,1] => 3 [6,4] => 4 [6,3,1] => 4 [6,2,2] => 4 [6,2,1,1] => 4 [6,1,1,1,1] => 4 [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] => 4 [4,4,2] => 6 [4,4,1,1] => 6 [4,3,3] => 6 [4,3,2,1] => 6 [4,3,1,1,1] => 5 [4,2,2,2] => 6 [4,2,2,1,1] => 5 [4,2,1,1,1,1] => 4 [4,1,1,1,1,1,1] => 3 [3,3,3,1] => 6 [3,3,2,2] => 6 [3,3,2,1,1] => 5 [3,3,1,1,1,1] => 4 [3,2,2,2,1] => 5 [3,2,2,1,1,1] => 4 [3,2,1,1,1,1,1] => 3 [3,1,1,1,1,1,1,1] => 2 [2,2,2,2,2] => 5 [2,2,2,2,1,1] => 4 [2,2,2,1,1,1,1] => 3 [2,2,1,1,1,1,1,1] => 2 [2,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1,1] => 0 [5,4,2] => 6 [5,4,1,1] => 6 [5,3,3] => 6 [5,3,2,1] => 6 [5,3,1,1,1] => 6 [5,2,2,2] => 6 [5,2,2,1,1] => 6 [4,4,3] => 7 [4,4,2,1] => 7 [4,4,1,1,1] => 6 [4,3,3,1] => 7 [4,3,2,2] => 7 [4,3,2,1,1] => 6 [4,2,2,2,1] => 6 [3,3,3,2] => 7 [3,3,3,1,1] => 6 [3,3,2,2,1] => 6 [6,4,2] => 6 [5,4,3] => 7 [5,4,2,1] => 7 [5,4,1,1,1] => 7 [5,3,3,1] => 7 [5,3,2,2] => 7 [5,3,2,1,1] => 7 [5,2,2,2,1] => 7 [4,4,3,1] => 8 [4,4,2,2] => 8 [4,4,2,1,1] => 7 [4,3,3,2] => 8 [4,3,3,1,1] => 7 [4,3,2,2,1] => 7 [3,3,3,2,1] => 7 [3,3,2,2,1,1] => 6 [5,4,3,1] => 8 [5,4,2,2] => 8 [5,4,2,1,1] => 8 [5,3,3,2] => 8 [5,3,3,1,1] => 8 [5,3,2,2,1] => 8 [4,4,3,2] => 9 [4,4,3,1,1] => 8 [4,4,2,2,1] => 8 [4,3,3,2,1] => 8 [5,4,3,2] => 9 [5,4,3,1,1] => 9 [5,4,2,2,1] => 9 [5,3,3,2,1] => 9 [4,4,3,2,1] => 9 [5,4,3,2,1] => 10 ----------------------------------------------------------------------------- Created: Jun 21, 2018 at 00:45 by Sara Billey ----------------------------------------------------------------------------- Last Updated: Jan 22, 2023 at 11:54 by Martin Rubey