***************************************************************************** * 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: St000377 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The dinv defect of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$. ----------------------------------------------------------------------------- References: [1] Lee, K., Li, L., Loehr, N. A. A Combinatorial Approach to the Symmetry of $q,t$-Catalan Numbers [[arXiv:1602.01126]] ----------------------------------------------------------------------------- Code: def statistic(P): return sum( 1 for c in P.cells() if P.arm_length(*c)-P.leg_length(*c) not in [0,1] ) ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => 0 [2] => 0 [1,1] => 1 [3] => 1 [2,1] => 0 [1,1,1] => 2 [4] => 2 [3,1] => 0 [2,2] => 1 [2,1,1] => 2 [1,1,1,1] => 3 [5] => 3 [4,1] => 2 [3,2] => 0 [3,1,1] => 1 [2,2,1] => 2 [2,1,1,1] => 3 [1,1,1,1,1] => 4 [6] => 4 [5,1] => 3 [4,2] => 1 [4,1,1] => 2 [3,3] => 2 [3,2,1] => 0 [3,1,1,1] => 3 [2,2,2] => 3 [2,2,1,1] => 4 [2,1,1,1,1] => 4 [1,1,1,1,1,1] => 5 [7] => 5 [6,1] => 4 [5,2] => 3 [5,1,1] => 4 [4,3] => 2 [4,2,1] => 0 [4,1,1,1] => 3 [3,3,1] => 1 [3,2,2] => 2 [3,2,1,1] => 3 [3,1,1,1,1] => 4 [2,2,2,1] => 4 [2,2,1,1,1] => 5 [2,1,1,1,1,1] => 5 [1,1,1,1,1,1,1] => 6 [8] => 6 [7,1] => 5 [6,2] => 4 [6,1,1] => 5 [5,3] => 3 [5,2,1] => 3 [5,1,1,1] => 4 [4,4] => 4 [4,3,1] => 0 [4,2,2] => 1 [4,2,1,1] => 2 [4,1,1,1,1] => 5 [3,3,2] => 2 [3,3,1,1] => 3 [3,2,2,1] => 4 [3,2,1,1,1] => 4 [3,1,1,1,1,1] => 5 [2,2,2,2] => 5 [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] => 7 [9] => 7 [8,1] => 6 [7,2] => 5 [7,1,1] => 6 [6,3] => 5 [6,2,1] => 4 [6,1,1,1] => 6 [5,4] => 4 [5,3,1] => 2 [5,2,2] => 3 [5,2,1,1] => 4 [5,1,1,1,1] => 5 [4,4,1] => 3 [4,3,2] => 0 [4,3,1,1] => 1 [4,2,2,1] => 2 [4,2,1,1,1] => 4 [4,1,1,1,1,1] => 6 [3,3,3] => 4 [3,3,2,1] => 3 [3,3,1,1,1] => 5 [3,2,2,2] => 5 [3,2,2,1,1] => 6 [3,2,1,1,1,1] => 5 [3,1,1,1,1,1,1] => 6 [2,2,2,2,1] => 6 [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] => 8 [10] => 8 [9,1] => 7 [8,2] => 6 [8,1,1] => 7 [7,3] => 6 [7,2,1] => 5 [7,1,1,1] => 7 [6,4] => 5 [6,3,1] => 4 [6,2,2] => 5 [6,2,1,1] => 6 [6,1,1,1,1] => 6 [5,5] => 6 [5,4,1] => 4 [5,3,2] => 1 [5,3,1,1] => 2 [5,2,2,1] => 3 [5,2,1,1,1] => 5 [5,1,1,1,1,1] => 7 [4,4,2] => 2 [4,4,1,1] => 3 [4,3,3] => 3 [4,3,2,1] => 0 [4,3,1,1,1] => 4 [4,2,2,2] => 4 [4,2,2,1,1] => 5 [4,2,1,1,1,1] => 5 [4,1,1,1,1,1,1] => 7 [3,3,3,1] => 4 [3,3,2,2] => 5 [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] => 7 [3,2,1,1,1,1,1] => 6 [3,1,1,1,1,1,1,1] => 7 [2,2,2,2,2] => 7 [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] => 9 [11] => 9 [10,1] => 8 [9,2] => 7 [9,1,1] => 8 [8,3] => 7 [8,2,1] => 6 [8,1,1,1] => 8 [7,4] => 7 [7,3,1] => 5 [7,2,2] => 6 [7,2,1,1] => 7 [7,1,1,1,1] => 8 [6,5] => 6 [6,4,1] => 5 [6,3,2] => 4 [6,3,1,1] => 5 [6,2,2,1] => 6 [6,2,1,1,1] => 6 [6,1,1,1,1,1] => 7 [5,5,1] => 6 [5,4,2] => 2 [5,4,1,1] => 3 [5,3,3] => 3 [5,3,2,1] => 0 [5,3,1,1,1] => 4 [5,2,2,2] => 4 [5,2,2,1,1] => 5 [5,2,1,1,1,1] => 7 [5,1,1,1,1,1,1] => 8 [4,4,3] => 4 [4,4,2,1] => 1 [4,4,1,1,1] => 5 [4,3,3,1] => 2 [4,3,2,2] => 3 [4,3,2,1,1] => 4 [4,3,1,1,1,1] => 5 [4,2,2,2,1] => 6 [4,2,2,1,1,1] => 6 [4,2,1,1,1,1,1] => 6 [4,1,1,1,1,1,1,1] => 8 [3,3,3,2] => 5 [3,3,3,1,1] => 6 [3,3,2,2,1] => 7 [3,3,2,1,1,1] => 7 [3,3,1,1,1,1,1] => 7 [3,2,2,2,2] => 7 [3,2,2,2,1,1] => 8 [3,2,2,1,1,1,1] => 8 [3,2,1,1,1,1,1,1] => 7 [3,1,1,1,1,1,1,1,1] => 8 [2,2,2,2,2,1] => 8 [2,2,2,2,1,1,1] => 9 [2,2,2,1,1,1,1,1] => 9 [2,2,1,1,1,1,1,1,1] => 9 [2,1,1,1,1,1,1,1,1,1] => 9 [1,1,1,1,1,1,1,1,1,1,1] => 10 [12] => 10 [11,1] => 9 [10,2] => 8 [10,1,1] => 9 [9,3] => 8 [9,2,1] => 7 [9,1,1,1] => 9 [8,4] => 8 [8,3,1] => 6 [8,2,2] => 7 [8,2,1,1] => 8 [8,1,1,1,1] => 9 [7,5] => 7 [7,4,1] => 7 [7,3,2] => 5 [7,3,1,1] => 6 [7,2,2,1] => 7 [7,2,1,1,1] => 8 [7,1,1,1,1,1] => 8 [6,6] => 8 [6,5,1] => 6 [6,4,2] => 4 [6,4,1,1] => 5 [6,3,3] => 5 [6,3,2,1] => 4 [6,3,1,1,1] => 6 [6,2,2,2] => 6 [6,2,2,1,1] => 7 [6,2,1,1,1,1] => 7 [6,1,1,1,1,1,1] => 9 [5,5,2] => 5 [5,5,1,1] => 6 [5,4,3] => 3 [5,4,2,1] => 0 [5,4,1,1,1] => 4 [5,3,3,1] => 1 [5,3,2,2] => 2 [5,3,2,1,1] => 3 [5,3,1,1,1,1] => 6 [5,2,2,2,1] => 5 [5,2,2,1,1,1] => 7 [5,2,1,1,1,1,1] => 8 [5,1,1,1,1,1,1,1] => 9 [4,4,4] => 6 [4,4,3,1] => 2 [4,4,2,2] => 3 [4,4,2,1,1] => 4 [4,4,1,1,1,1] => 7 [4,3,3,2] => 4 [4,3,3,1,1] => 5 [4,3,2,2,1] => 6 [4,3,2,1,1,1] => 5 [4,3,1,1,1,1,1] => 6 [4,2,2,2,2] => 7 [4,2,2,2,1,1] => 8 [4,2,2,1,1,1,1] => 7 [4,2,1,1,1,1,1,1] => 7 [4,1,1,1,1,1,1,1,1] => 9 [3,3,3,3] => 7 [3,3,3,2,1] => 6 [3,3,3,1,1,1] => 8 [3,3,2,2,2] => 8 [3,3,2,2,1,1] => 9 [3,3,2,1,1,1,1] => 8 [3,3,1,1,1,1,1,1] => 8 [3,2,2,2,2,1] => 8 [3,2,2,2,1,1,1] => 9 [3,2,2,1,1,1,1,1] => 9 [3,2,1,1,1,1,1,1,1] => 8 [3,1,1,1,1,1,1,1,1,1] => 9 [2,2,2,2,2,2] => 9 [2,2,2,2,2,1,1] => 10 [2,2,2,2,1,1,1,1] => 10 [2,2,2,1,1,1,1,1,1] => 10 [2,2,1,1,1,1,1,1,1,1] => 10 [2,1,1,1,1,1,1,1,1,1,1] => 10 [1,1,1,1,1,1,1,1,1,1,1,1] => 11 [8,5] => 9 [7,5,1] => 7 [7,4,2] => 6 [5,5,3] => 5 [5,4,4] => 6 [5,4,3,1] => 0 [5,4,2,2] => 1 [5,4,2,1,1] => 2 [5,3,3,2] => 2 [5,3,3,1,1] => 3 [5,3,2,2,1] => 4 [4,4,4,1] => 5 [4,4,3,2] => 3 [4,4,3,1,1] => 4 [4,4,2,2,1] => 5 [4,3,3,3] => 6 [4,3,3,2,1] => 6 [3,3,3,3,1] => 7 [3,3,3,2,2] => 8 [9,5] => 10 [8,5,1] => 9 [7,5,2] => 7 [7,4,3] => 7 [5,5,4] => 7 [5,4,3,2] => 0 [5,4,3,1,1] => 1 [5,4,2,2,1] => 2 [5,3,3,2,1] => 3 [5,3,2,2,2] => 6 [4,4,4,2] => 5 [4,4,3,3] => 6 [4,4,3,2,1] => 4 [3,3,3,3,2] => 8 [9,5,1] => 10 [8,5,2] => 9 [7,5,3] => 7 [5,5,5] => 9 [5,4,3,2,1] => 0 [5,3,2,2,2,1] => 8 [4,4,4,3] => 7 [3,3,3,3,3] => 10 [8,5,3] => 9 [7,5,3,1] => 6 [4,4,4,4] => 9 [8,6,3] => 10 [9,6,3] => 12 [8,6,4] => 10 [9,6,4] => 12 [8,5,4,2] => 8 [8,5,5,1] => 11 [7,5,4,3,1] => 2 [8,6,4,2] => 9 [10,6,4] => 13 [10,7,3] => 14 [9,7,4] => 13 [9,5,5,1] => 13 [6,5,4,3,2,1] => 0 [11,7,3] => 15 [9,6,4,3] => 11 [9,6,5,3] => 12 [8,6,5,3,1] => 8 [11,7,5,1] => 16 [9,7,5,3] => 13 [9,7,5,3,1] => 12 [10,7,5,3] => 15 [9,7,5,4,1] => 13 [7,6,5,4,3,2,1] => 0 [10,7,6,4,1] => 16 [9,7,6,4,2] => 12 [10,8,5,4,1] => 17 [10,8,6,4,1] => 17 [9,7,5,5,3,1] => 10 [11,8,6,4,1] => 19 [10,8,6,4,2] => 16 [11,8,6,5,1] => 19 [12,9,7,5,1] => 23 [13,9,7,5,1] => 24 [11,9,7,5,3,1] => 20 [11,8,7,5,4,1] => 19 [8,7,6,5,4,3,2,1] => 0 [11,9,7,5,5,3] => 20 [11,9,7,7,5,3,3] => 18 [11,9,7,6,5,3,1] => 18 [13,11,9,7,5,3,1] => 30 [13,11,9,7,7,5,3,1] => 30 [17,13,11,9,7,5,1] => 46 [15,13,11,9,7,5,3,1] => 42 [29,23,19,17,13,11,7,1] => 103 ----------------------------------------------------------------------------- Created: Feb 06, 2016 at 17:12 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Feb 25, 2021 at 20:14 by Martin Rubey