***************************************************************************** * 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: St000378 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The diagonal inversion number of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$. ----------------------------------------------------------------------------- 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) in [0,1] ) ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => 1 [2] => 2 [1,1] => 1 [3] => 2 [2,1] => 3 [1,1,1] => 1 [4] => 2 [3,1] => 4 [2,2] => 3 [2,1,1] => 2 [1,1,1,1] => 1 [5] => 2 [4,1] => 3 [3,2] => 5 [3,1,1] => 4 [2,2,1] => 3 [2,1,1,1] => 2 [1,1,1,1,1] => 1 [6] => 2 [5,1] => 3 [4,2] => 5 [4,1,1] => 4 [3,3] => 4 [3,2,1] => 6 [3,1,1,1] => 3 [2,2,2] => 3 [2,2,1,1] => 2 [2,1,1,1,1] => 2 [1,1,1,1,1,1] => 1 [7] => 2 [6,1] => 3 [5,2] => 4 [5,1,1] => 3 [4,3] => 5 [4,2,1] => 7 [4,1,1,1] => 4 [3,3,1] => 6 [3,2,2] => 5 [3,2,1,1] => 4 [3,1,1,1,1] => 3 [2,2,2,1] => 3 [2,2,1,1,1] => 2 [2,1,1,1,1,1] => 2 [1,1,1,1,1,1,1] => 1 [8] => 2 [7,1] => 3 [6,2] => 4 [6,1,1] => 3 [5,3] => 5 [5,2,1] => 5 [5,1,1,1] => 4 [4,4] => 4 [4,3,1] => 8 [4,2,2] => 7 [4,2,1,1] => 6 [4,1,1,1,1] => 3 [3,3,2] => 6 [3,3,1,1] => 5 [3,2,2,1] => 4 [3,2,1,1,1] => 4 [3,1,1,1,1,1] => 3 [2,2,2,2] => 3 [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 [9] => 2 [8,1] => 3 [7,2] => 4 [7,1,1] => 3 [6,3] => 4 [6,2,1] => 5 [6,1,1,1] => 3 [5,4] => 5 [5,3,1] => 7 [5,2,2] => 6 [5,2,1,1] => 5 [5,1,1,1,1] => 4 [4,4,1] => 6 [4,3,2] => 9 [4,3,1,1] => 8 [4,2,2,1] => 7 [4,2,1,1,1] => 5 [4,1,1,1,1,1] => 3 [3,3,3] => 5 [3,3,2,1] => 6 [3,3,1,1,1] => 4 [3,2,2,2] => 4 [3,2,2,1,1] => 3 [3,2,1,1,1,1] => 4 [3,1,1,1,1,1,1] => 3 [2,2,2,2,1] => 3 [2,2,2,1,1,1] => 2 [2,2,1,1,1,1,1] => 2 [2,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1] => 1 [10] => 2 [9,1] => 3 [8,2] => 4 [8,1,1] => 3 [7,3] => 4 [7,2,1] => 5 [7,1,1,1] => 3 [6,4] => 5 [6,3,1] => 6 [6,2,2] => 5 [6,2,1,1] => 4 [6,1,1,1,1] => 4 [5,5] => 4 [5,4,1] => 6 [5,3,2] => 9 [5,3,1,1] => 8 [5,2,2,1] => 7 [5,2,1,1,1] => 5 [5,1,1,1,1,1] => 3 [4,4,2] => 8 [4,4,1,1] => 7 [4,3,3] => 7 [4,3,2,1] => 10 [4,3,1,1,1] => 6 [4,2,2,2] => 6 [4,2,2,1,1] => 5 [4,2,1,1,1,1] => 5 [4,1,1,1,1,1,1] => 3 [3,3,3,1] => 6 [3,3,2,2] => 5 [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] => 3 [3,2,1,1,1,1,1] => 4 [3,1,1,1,1,1,1,1] => 3 [2,2,2,2,2] => 3 [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 [11] => 2 [10,1] => 3 [9,2] => 4 [9,1,1] => 3 [8,3] => 4 [8,2,1] => 5 [8,1,1,1] => 3 [7,4] => 4 [7,3,1] => 6 [7,2,2] => 5 [7,2,1,1] => 4 [7,1,1,1,1] => 3 [6,5] => 5 [6,4,1] => 6 [6,3,2] => 7 [6,3,1,1] => 6 [6,2,2,1] => 5 [6,2,1,1,1] => 5 [6,1,1,1,1,1] => 4 [5,5,1] => 5 [5,4,2] => 9 [5,4,1,1] => 8 [5,3,3] => 8 [5,3,2,1] => 11 [5,3,1,1,1] => 7 [5,2,2,2] => 7 [5,2,2,1,1] => 6 [5,2,1,1,1,1] => 4 [5,1,1,1,1,1,1] => 3 [4,4,3] => 7 [4,4,2,1] => 10 [4,4,1,1,1] => 6 [4,3,3,1] => 9 [4,3,2,2] => 8 [4,3,2,1,1] => 7 [4,3,1,1,1,1] => 6 [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] => 3 [3,3,3,2] => 6 [3,3,3,1,1] => 5 [3,3,2,2,1] => 4 [3,3,2,1,1,1] => 4 [3,3,1,1,1,1,1] => 4 [3,2,2,2,2] => 4 [3,2,2,2,1,1] => 3 [3,2,2,1,1,1,1] => 3 [3,2,1,1,1,1,1,1] => 4 [3,1,1,1,1,1,1,1,1] => 3 [2,2,2,2,2,1] => 3 [2,2,2,2,1,1,1] => 2 [2,2,2,1,1,1,1,1] => 2 [2,2,1,1,1,1,1,1,1] => 2 [2,1,1,1,1,1,1,1,1,1] => 2 [1,1,1,1,1,1,1,1,1,1,1] => 1 [12] => 2 [11,1] => 3 [10,2] => 4 [10,1,1] => 3 [9,3] => 4 [9,2,1] => 5 [9,1,1,1] => 3 [8,4] => 4 [8,3,1] => 6 [8,2,2] => 5 [8,2,1,1] => 4 [8,1,1,1,1] => 3 [7,5] => 5 [7,4,1] => 5 [7,3,2] => 7 [7,3,1,1] => 6 [7,2,2,1] => 5 [7,2,1,1,1] => 4 [7,1,1,1,1,1] => 4 [6,6] => 4 [6,5,1] => 6 [6,4,2] => 8 [6,4,1,1] => 7 [6,3,3] => 7 [6,3,2,1] => 8 [6,3,1,1,1] => 6 [6,2,2,2] => 6 [6,2,2,1,1] => 5 [6,2,1,1,1,1] => 5 [6,1,1,1,1,1,1] => 3 [5,5,2] => 7 [5,5,1,1] => 6 [5,4,3] => 9 [5,4,2,1] => 12 [5,4,1,1,1] => 8 [5,3,3,1] => 11 [5,3,2,2] => 10 [5,3,2,1,1] => 9 [5,3,1,1,1,1] => 6 [5,2,2,2,1] => 7 [5,2,2,1,1,1] => 5 [5,2,1,1,1,1,1] => 4 [5,1,1,1,1,1,1,1] => 3 [4,4,4] => 6 [4,4,3,1] => 10 [4,4,2,2] => 9 [4,4,2,1,1] => 8 [4,4,1,1,1,1] => 5 [4,3,3,2] => 8 [4,3,3,1,1] => 7 [4,3,2,2,1] => 6 [4,3,2,1,1,1] => 7 [4,3,1,1,1,1,1] => 6 [4,2,2,2,2] => 5 [4,2,2,2,1,1] => 4 [4,2,2,1,1,1,1] => 5 [4,2,1,1,1,1,1,1] => 5 [4,1,1,1,1,1,1,1,1] => 3 [3,3,3,3] => 5 [3,3,3,2,1] => 6 [3,3,3,1,1,1] => 4 [3,3,2,2,2] => 4 [3,3,2,2,1,1] => 3 [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] => 3 [3,2,2,1,1,1,1,1] => 3 [3,2,1,1,1,1,1,1,1] => 4 [3,1,1,1,1,1,1,1,1,1] => 3 [2,2,2,2,2,2] => 3 [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 ----------------------------------------------------------------------------- Created: Feb 06, 2016 at 17:13 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Oct 31, 2017 at 08:15 by Martin Rubey