***************************************************************************** * 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: St000566 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The number of ways to select a row of a Ferrers shape and two cells in this row. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(pi): return sum(binomial(p, Integer(2)) for p in pi) ----------------------------------------------------------------------------- Statistic values: [2] => 1 [1,1] => 0 [3] => 3 [2,1] => 1 [1,1,1] => 0 [4] => 6 [3,1] => 3 [2,2] => 2 [2,1,1] => 1 [1,1,1,1] => 0 [5] => 10 [4,1] => 6 [3,2] => 4 [3,1,1] => 3 [2,2,1] => 2 [2,1,1,1] => 1 [1,1,1,1,1] => 0 [6] => 15 [5,1] => 10 [4,2] => 7 [4,1,1] => 6 [3,3] => 6 [3,2,1] => 4 [3,1,1,1] => 3 [2,2,2] => 3 [2,2,1,1] => 2 [2,1,1,1,1] => 1 [1,1,1,1,1,1] => 0 [7] => 21 [6,1] => 15 [5,2] => 11 [5,1,1] => 10 [4,3] => 9 [4,2,1] => 7 [4,1,1,1] => 6 [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] => 1 [1,1,1,1,1,1,1] => 0 [8] => 28 [7,1] => 21 [6,2] => 16 [6,1,1] => 15 [5,3] => 13 [5,2,1] => 11 [5,1,1,1] => 10 [4,4] => 12 [4,3,1] => 9 [4,2,2] => 8 [4,2,1,1] => 7 [4,1,1,1,1] => 6 [3,3,2] => 7 [3,3,1,1] => 6 [3,2,2,1] => 5 [3,2,1,1,1] => 4 [3,1,1,1,1,1] => 3 [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] => 36 [8,1] => 28 [7,2] => 22 [7,1,1] => 21 [6,3] => 18 [6,2,1] => 16 [6,1,1,1] => 15 [5,4] => 16 [5,3,1] => 13 [5,2,2] => 12 [5,2,1,1] => 11 [5,1,1,1,1] => 10 [4,4,1] => 12 [4,3,2] => 10 [4,3,1,1] => 9 [4,2,2,1] => 8 [4,2,1,1,1] => 7 [4,1,1,1,1,1] => 6 [3,3,3] => 9 [3,3,2,1] => 7 [3,3,1,1,1] => 6 [3,2,2,2] => 6 [3,2,2,1,1] => 5 [3,2,1,1,1,1] => 4 [3,1,1,1,1,1,1] => 3 [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] => 45 [9,1] => 36 [8,2] => 29 [8,1,1] => 28 [7,3] => 24 [7,2,1] => 22 [7,1,1,1] => 21 [6,4] => 21 [6,3,1] => 18 [6,2,2] => 17 [6,2,1,1] => 16 [6,1,1,1,1] => 15 [5,5] => 20 [5,4,1] => 16 [5,3,2] => 14 [5,3,1,1] => 13 [5,2,2,1] => 12 [5,2,1,1,1] => 11 [5,1,1,1,1,1] => 10 [4,4,2] => 13 [4,4,1,1] => 12 [4,3,3] => 12 [4,3,2,1] => 10 [4,3,1,1,1] => 9 [4,2,2,2] => 9 [4,2,2,1,1] => 8 [4,2,1,1,1,1] => 7 [4,1,1,1,1,1,1] => 6 [3,3,3,1] => 9 [3,3,2,2] => 8 [3,3,2,1,1] => 7 [3,3,1,1,1,1] => 6 [3,2,2,2,1] => 6 [3,2,2,1,1,1] => 5 [3,2,1,1,1,1,1] => 4 [3,1,1,1,1,1,1,1] => 3 [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 [11] => 55 [10,1] => 45 [9,2] => 37 [9,1,1] => 36 [8,3] => 31 [8,2,1] => 29 [8,1,1,1] => 28 [7,4] => 27 [7,3,1] => 24 [7,2,2] => 23 [7,2,1,1] => 22 [7,1,1,1,1] => 21 [6,5] => 25 [6,4,1] => 21 [6,3,2] => 19 [6,3,1,1] => 18 [6,2,2,1] => 17 [6,2,1,1,1] => 16 [6,1,1,1,1,1] => 15 [5,5,1] => 20 [5,4,2] => 17 [5,4,1,1] => 16 [5,3,3] => 16 [5,3,2,1] => 14 [5,3,1,1,1] => 13 [5,2,2,2] => 13 [5,2,2,1,1] => 12 [5,2,1,1,1,1] => 11 [5,1,1,1,1,1,1] => 10 [4,4,3] => 15 [4,4,2,1] => 13 [4,4,1,1,1] => 12 [4,3,3,1] => 12 [4,3,2,2] => 11 [4,3,2,1,1] => 10 [4,3,1,1,1,1] => 9 [4,2,2,2,1] => 9 [4,2,2,1,1,1] => 8 [4,2,1,1,1,1,1] => 7 [4,1,1,1,1,1,1,1] => 6 [3,3,3,2] => 10 [3,3,3,1,1] => 9 [3,3,2,2,1] => 8 [3,3,2,1,1,1] => 7 [3,3,1,1,1,1,1] => 6 [3,2,2,2,2] => 7 [3,2,2,2,1,1] => 6 [3,2,2,1,1,1,1] => 5 [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] => 5 [2,2,2,2,1,1,1] => 4 [2,2,2,1,1,1,1,1] => 3 [2,2,1,1,1,1,1,1,1] => 2 [2,1,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1,1,1] => 0 [12] => 66 [11,1] => 55 [10,2] => 46 [10,1,1] => 45 [9,3] => 39 [9,2,1] => 37 [9,1,1,1] => 36 [8,4] => 34 [8,3,1] => 31 [8,2,2] => 30 [8,2,1,1] => 29 [8,1,1,1,1] => 28 [7,5] => 31 [7,4,1] => 27 [7,3,2] => 25 [7,3,1,1] => 24 [7,2,2,1] => 23 [7,2,1,1,1] => 22 [7,1,1,1,1,1] => 21 [6,6] => 30 [6,5,1] => 25 [6,4,2] => 22 [6,4,1,1] => 21 [6,3,3] => 21 [6,3,2,1] => 19 [6,3,1,1,1] => 18 [6,2,2,2] => 18 [6,2,2,1,1] => 17 [6,2,1,1,1,1] => 16 [6,1,1,1,1,1,1] => 15 [5,5,2] => 21 [5,5,1,1] => 20 [5,4,3] => 19 [5,4,2,1] => 17 [5,4,1,1,1] => 16 [5,3,3,1] => 16 [5,3,2,2] => 15 [5,3,2,1,1] => 14 [5,3,1,1,1,1] => 13 [5,2,2,2,1] => 13 [5,2,2,1,1,1] => 12 [5,2,1,1,1,1,1] => 11 [5,1,1,1,1,1,1,1] => 10 [4,4,4] => 18 [4,4,3,1] => 15 [4,4,2,2] => 14 [4,4,2,1,1] => 13 [4,4,1,1,1,1] => 12 [4,3,3,2] => 13 [4,3,3,1,1] => 12 [4,3,2,2,1] => 11 [4,3,2,1,1,1] => 10 [4,3,1,1,1,1,1] => 9 [4,2,2,2,2] => 10 [4,2,2,2,1,1] => 9 [4,2,2,1,1,1,1] => 8 [4,2,1,1,1,1,1,1] => 7 [4,1,1,1,1,1,1,1,1] => 6 [3,3,3,3] => 12 [3,3,3,2,1] => 10 [3,3,3,1,1,1] => 9 [3,3,2,2,2] => 9 [3,3,2,2,1,1] => 8 [3,3,2,1,1,1,1] => 7 [3,3,1,1,1,1,1,1] => 6 [3,2,2,2,2,1] => 7 [3,2,2,2,1,1,1] => 6 [3,2,2,1,1,1,1,1] => 5 [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] => 6 [2,2,2,2,2,1,1] => 5 [2,2,2,2,1,1,1,1] => 4 [2,2,2,1,1,1,1,1,1] => 3 [2,2,1,1,1,1,1,1,1,1] => 2 [2,1,1,1,1,1,1,1,1,1,1] => 1 [1,1,1,1,1,1,1,1,1,1,1,1] => 0 ----------------------------------------------------------------------------- Created: Aug 07, 2016 at 13:27 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Aug 07, 2016 at 13:27 by Martin Rubey