***************************************************************************** * 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: St000509 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The diagonal index (content) of a partition. The '''diagonal index''' of the cell at row $r$ and column $c$ of a partition is $c - r$; this is sometimes called the '''content''' of the cell. The '''diagonal index of a partition''' is the sum of the diagonal index of each cell of the partition. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(la): return sum((j - i) for (i, j) in la.cells()) ----------------------------------------------------------------------------- Statistic values: [2] => 1 [1,1] => -1 [3] => 3 [2,1] => 0 [1,1,1] => -3 [4] => 6 [3,1] => 2 [2,2] => 0 [2,1,1] => -2 [1,1,1,1] => -6 [5] => 10 [4,1] => 5 [3,2] => 2 [3,1,1] => 0 [2,2,1] => -2 [2,1,1,1] => -5 [1,1,1,1,1] => -10 [6] => 15 [5,1] => 9 [4,2] => 5 [4,1,1] => 3 [3,3] => 3 [3,2,1] => 0 [3,1,1,1] => -3 [2,2,2] => -3 [2,2,1,1] => -5 [2,1,1,1,1] => -9 [1,1,1,1,1,1] => -15 [7] => 21 [6,1] => 14 [5,2] => 9 [5,1,1] => 7 [4,3] => 6 [4,2,1] => 3 [4,1,1,1] => 0 [3,3,1] => 1 [3,2,2] => -1 [3,2,1,1] => -3 [3,1,1,1,1] => -7 [2,2,2,1] => -6 [2,2,1,1,1] => -9 [2,1,1,1,1,1] => -14 [1,1,1,1,1,1,1] => -21 [8] => 28 [7,1] => 20 [6,2] => 14 [6,1,1] => 12 [5,3] => 10 [5,2,1] => 7 [5,1,1,1] => 4 [4,4] => 8 [4,3,1] => 4 [4,2,2] => 2 [4,2,1,1] => 0 [4,1,1,1,1] => -4 [3,3,2] => 0 [3,3,1,1] => -2 [3,2,2,1] => -4 [3,2,1,1,1] => -7 [3,1,1,1,1,1] => -12 [2,2,2,2] => -8 [2,2,2,1,1] => -10 [2,2,1,1,1,1] => -14 [2,1,1,1,1,1,1] => -20 [1,1,1,1,1,1,1,1] => -28 [9] => 36 [8,1] => 27 [7,2] => 20 [7,1,1] => 18 [6,3] => 15 [6,2,1] => 12 [6,1,1,1] => 9 [5,4] => 12 [5,3,1] => 8 [5,2,2] => 6 [5,2,1,1] => 4 [5,1,1,1,1] => 0 [4,4,1] => 6 [4,3,2] => 3 [4,3,1,1] => 1 [4,2,2,1] => -1 [4,2,1,1,1] => -4 [4,1,1,1,1,1] => -9 [3,3,3] => 0 [3,3,2,1] => -3 [3,3,1,1,1] => -6 [3,2,2,2] => -6 [3,2,2,1,1] => -8 [3,2,1,1,1,1] => -12 [3,1,1,1,1,1,1] => -18 [2,2,2,2,1] => -12 [2,2,2,1,1,1] => -15 [2,2,1,1,1,1,1] => -20 [2,1,1,1,1,1,1,1] => -27 [1,1,1,1,1,1,1,1,1] => -36 [10] => 45 [9,1] => 35 [8,2] => 27 [8,1,1] => 25 [7,3] => 21 [7,2,1] => 18 [7,1,1,1] => 15 [6,4] => 17 [6,3,1] => 13 [6,2,2] => 11 [6,2,1,1] => 9 [6,1,1,1,1] => 5 [5,5] => 15 [5,4,1] => 10 [5,3,2] => 7 [5,3,1,1] => 5 [5,2,2,1] => 3 [5,2,1,1,1] => 0 [5,1,1,1,1,1] => -5 [4,4,2] => 5 [4,4,1,1] => 3 [4,3,3] => 3 [4,3,2,1] => 0 [4,3,1,1,1] => -3 [4,2,2,2] => -3 [4,2,2,1,1] => -5 [4,2,1,1,1,1] => -9 [4,1,1,1,1,1,1] => -15 [3,3,3,1] => -3 [3,3,2,2] => -5 [3,3,2,1,1] => -7 [3,3,1,1,1,1] => -11 [3,2,2,2,1] => -10 [3,2,2,1,1,1] => -13 [3,2,1,1,1,1,1] => -18 [3,1,1,1,1,1,1,1] => -25 [2,2,2,2,2] => -15 [2,2,2,2,1,1] => -17 [2,2,2,1,1,1,1] => -21 [2,2,1,1,1,1,1,1] => -27 [2,1,1,1,1,1,1,1,1] => -35 [1,1,1,1,1,1,1,1,1,1] => -45 [11] => 55 [10,1] => 44 [9,2] => 35 [9,1,1] => 33 [8,3] => 28 [8,2,1] => 25 [8,1,1,1] => 22 [7,4] => 23 [7,3,1] => 19 [7,2,2] => 17 [7,2,1,1] => 15 [7,1,1,1,1] => 11 [6,5] => 20 [6,4,1] => 15 [6,3,2] => 12 [6,3,1,1] => 10 [6,2,2,1] => 8 [6,2,1,1,1] => 5 [6,1,1,1,1,1] => 0 [5,5,1] => 13 [5,4,2] => 9 [5,4,1,1] => 7 [5,3,3] => 7 [5,3,2,1] => 4 [5,3,1,1,1] => 1 [5,2,2,2] => 1 [5,2,2,1,1] => -1 [5,2,1,1,1,1] => -5 [5,1,1,1,1,1,1] => -11 [4,4,3] => 5 [4,4,2,1] => 2 [4,4,1,1,1] => -1 [4,3,3,1] => 0 [4,3,2,2] => -2 [4,3,2,1,1] => -4 [4,3,1,1,1,1] => -8 [4,2,2,2,1] => -7 [4,2,2,1,1,1] => -10 [4,2,1,1,1,1,1] => -15 [4,1,1,1,1,1,1,1] => -22 [3,3,3,2] => -5 [3,3,3,1,1] => -7 [3,3,2,2,1] => -9 [3,3,2,1,1,1] => -12 [3,3,1,1,1,1,1] => -17 [3,2,2,2,2] => -13 [3,2,2,2,1,1] => -15 [3,2,2,1,1,1,1] => -19 [3,2,1,1,1,1,1,1] => -25 [3,1,1,1,1,1,1,1,1] => -33 [2,2,2,2,2,1] => -20 [2,2,2,2,1,1,1] => -23 [2,2,2,1,1,1,1,1] => -28 [2,2,1,1,1,1,1,1,1] => -35 [2,1,1,1,1,1,1,1,1,1] => -44 [1,1,1,1,1,1,1,1,1,1,1] => -55 [12] => 66 [11,1] => 54 [10,2] => 44 [10,1,1] => 42 [9,3] => 36 [9,2,1] => 33 [9,1,1,1] => 30 [8,4] => 30 [8,3,1] => 26 [8,2,2] => 24 [8,2,1,1] => 22 [8,1,1,1,1] => 18 [7,5] => 26 [7,4,1] => 21 [7,3,2] => 18 [7,3,1,1] => 16 [7,2,2,1] => 14 [7,2,1,1,1] => 11 [7,1,1,1,1,1] => 6 [6,6] => 24 [6,5,1] => 18 [6,4,2] => 14 [6,4,1,1] => 12 [6,3,3] => 12 [6,3,2,1] => 9 [6,3,1,1,1] => 6 [6,2,2,2] => 6 [6,2,2,1,1] => 4 [6,2,1,1,1,1] => 0 [6,1,1,1,1,1,1] => -6 [5,5,2] => 12 [5,5,1,1] => 10 [5,4,3] => 9 [5,4,2,1] => 6 [5,4,1,1,1] => 3 [5,3,3,1] => 4 [5,3,2,2] => 2 [5,3,2,1,1] => 0 [5,3,1,1,1,1] => -4 [5,2,2,2,1] => -3 [5,2,2,1,1,1] => -6 [5,2,1,1,1,1,1] => -11 [5,1,1,1,1,1,1,1] => -18 [4,4,4] => 6 [4,4,3,1] => 2 [4,4,2,2] => 0 [4,4,2,1,1] => -2 [4,4,1,1,1,1] => -6 [4,3,3,2] => -2 [4,3,3,1,1] => -4 [4,3,2,2,1] => -6 [4,3,2,1,1,1] => -9 [4,3,1,1,1,1,1] => -14 [4,2,2,2,2] => -10 [4,2,2,2,1,1] => -12 [4,2,2,1,1,1,1] => -16 [4,2,1,1,1,1,1,1] => -22 [4,1,1,1,1,1,1,1,1] => -30 [3,3,3,3] => -6 [3,3,3,2,1] => -9 [3,3,3,1,1,1] => -12 [3,3,2,2,2] => -12 [3,3,2,2,1,1] => -14 [3,3,2,1,1,1,1] => -18 [3,3,1,1,1,1,1,1] => -24 [3,2,2,2,2,1] => -18 [3,2,2,2,1,1,1] => -21 [3,2,2,1,1,1,1,1] => -26 [3,2,1,1,1,1,1,1,1] => -33 [3,1,1,1,1,1,1,1,1,1] => -42 [2,2,2,2,2,2] => -24 [2,2,2,2,2,1,1] => -26 [2,2,2,2,1,1,1,1] => -30 [2,2,2,1,1,1,1,1,1] => -36 [2,2,1,1,1,1,1,1,1,1] => -44 [2,1,1,1,1,1,1,1,1,1,1] => -54 [1,1,1,1,1,1,1,1,1,1,1,1] => -66 ----------------------------------------------------------------------------- Created: May 25, 2016 at 20:28 by Franco Saliola ----------------------------------------------------------------------------- Last Updated: May 25, 2016 at 20:28 by Franco Saliola