***************************************************************************** * 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: St000008 ----------------------------------------------------------------------------- Collection: Integer compositions ----------------------------------------------------------------------------- Description: The major index of the composition. The descents of a composition \$[c_1,c_2,\dots,c_k]\$ are the partial sums \$c_1, c_1+c_2,\dots, c_1+\dots+c_{k-1}\$, excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]]. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(x): return x.major_index() ----------------------------------------------------------------------------- Statistic values: [1] => 0 [1,1] => 1 [2] => 0 [1,1,1] => 3 [1,2] => 1 [2,1] => 2 [3] => 0 [1,1,1,1] => 6 [1,1,2] => 3 [1,2,1] => 4 [1,3] => 1 [2,1,1] => 5 [2,2] => 2 [3,1] => 3 [4] => 0 [1,1,1,1,1] => 10 [1,1,1,2] => 6 [1,1,2,1] => 7 [1,1,3] => 3 [1,2,1,1] => 8 [1,2,2] => 4 [1,3,1] => 5 [1,4] => 1 [2,1,1,1] => 9 [2,1,2] => 5 [2,2,1] => 6 [2,3] => 2 [3,1,1] => 7 [3,2] => 3 [4,1] => 4 [5] => 0 [1,1,1,1,1,1] => 15 [1,1,1,1,2] => 10 [1,1,1,2,1] => 11 [1,1,1,3] => 6 [1,1,2,1,1] => 12 [1,1,2,2] => 7 [1,1,3,1] => 8 [1,1,4] => 3 [1,2,1,1,1] => 13 [1,2,1,2] => 8 [1,2,2,1] => 9 [1,2,3] => 4 [1,3,1,1] => 10 [1,3,2] => 5 [1,4,1] => 6 [1,5] => 1 [2,1,1,1,1] => 14 [2,1,1,2] => 9 [2,1,2,1] => 10 [2,1,3] => 5 [2,2,1,1] => 11 [2,2,2] => 6 [2,3,1] => 7 [2,4] => 2 [3,1,1,1] => 12 [3,1,2] => 7 [3,2,1] => 8 [3,3] => 3 [4,1,1] => 9 [4,2] => 4 [5,1] => 5 [6] => 0 [1,1,1,1,1,1,1] => 21 [1,1,1,1,1,2] => 15 [1,1,1,1,2,1] => 16 [1,1,1,1,3] => 10 [1,1,1,2,1,1] => 17 [1,1,1,2,2] => 11 [1,1,1,3,1] => 12 [1,1,1,4] => 6 [1,1,2,1,1,1] => 18 [1,1,2,1,2] => 12 [1,1,2,2,1] => 13 [1,1,2,3] => 7 [1,1,3,1,1] => 14 [1,1,3,2] => 8 [1,1,4,1] => 9 [1,1,5] => 3 [1,2,1,1,1,1] => 19 [1,2,1,1,2] => 13 [1,2,1,2,1] => 14 [1,2,1,3] => 8 [1,2,2,1,1] => 15 [1,2,2,2] => 9 [1,2,3,1] => 10 [1,2,4] => 4 [1,3,1,1,1] => 16 [1,3,1,2] => 10 [1,3,2,1] => 11 [1,3,3] => 5 [1,4,1,1] => 12 [1,4,2] => 6 [1,5,1] => 7 [1,6] => 1 [2,1,1,1,1,1] => 20 [2,1,1,1,2] => 14 [2,1,1,2,1] => 15 [2,1,1,3] => 9 [2,1,2,1,1] => 16 [2,1,2,2] => 10 [2,1,3,1] => 11 [2,1,4] => 5 [2,2,1,1,1] => 17 [2,2,1,2] => 11 [2,2,2,1] => 12 [2,2,3] => 6 [2,3,1,1] => 13 [2,3,2] => 7 [2,4,1] => 8 [2,5] => 2 [3,1,1,1,1] => 18 [3,1,1,2] => 12 [3,1,2,1] => 13 [3,1,3] => 7 [3,2,1,1] => 14 [3,2,2] => 8 [3,3,1] => 9 [3,4] => 3 [4,1,1,1] => 15 [4,1,2] => 9 [4,2,1] => 10 [4,3] => 4 [5,1,1] => 11 [5,2] => 5 [6,1] => 6 [7] => 0 [1,1,1,1,1,1,1,1] => 28 [1,1,1,1,1,1,2] => 21 [1,1,1,1,1,2,1] => 22 [1,1,1,1,1,3] => 15 [1,1,1,1,2,1,1] => 23 [1,1,1,1,2,2] => 16 [1,1,1,1,3,1] => 17 [1,1,1,1,4] => 10 [1,1,1,2,1,1,1] => 24 [1,1,1,2,1,2] => 17 [1,1,1,2,2,1] => 18 [1,1,1,2,3] => 11 [1,1,1,3,1,1] => 19 [1,1,1,3,2] => 12 [1,1,1,4,1] => 13 [1,1,1,5] => 6 [1,1,2,1,1,1,1] => 25 [1,1,2,1,1,2] => 18 [1,1,2,1,2,1] => 19 [1,1,2,1,3] => 12 [1,1,2,2,1,1] => 20 [1,1,2,2,2] => 13 [1,1,2,3,1] => 14 [1,1,2,4] => 7 [1,1,3,1,1,1] => 21 [1,1,3,1,2] => 14 [1,1,3,2,1] => 15 [1,1,3,3] => 8 [1,1,4,1,1] => 16 [1,1,4,2] => 9 [1,1,5,1] => 10 [1,1,6] => 3 [1,2,1,1,1,1,1] => 26 [1,2,1,1,1,2] => 19 [1,2,1,1,2,1] => 20 [1,2,1,1,3] => 13 [1,2,1,2,1,1] => 21 [1,2,1,2,2] => 14 [1,2,1,3,1] => 15 [1,2,1,4] => 8 [1,2,2,1,1,1] => 22 [1,2,2,1,2] => 15 [1,2,2,2,1] => 16 [1,2,2,3] => 9 [1,2,3,1,1] => 17 [1,2,3,2] => 10 [1,2,4,1] => 11 [1,2,5] => 4 [1,3,1,1,1,1] => 23 [1,3,1,1,2] => 16 [1,3,1,2,1] => 17 [1,3,1,3] => 10 [1,3,2,1,1] => 18 [1,3,2,2] => 11 [1,3,3,1] => 12 [1,3,4] => 5 [1,4,1,1,1] => 19 [1,4,1,2] => 12 [1,4,2,1] => 13 [1,4,3] => 6 [1,5,1,1] => 14 [1,5,2] => 7 [1,6,1] => 8 [1,7] => 1 [2,1,1,1,1,1,1] => 27 [2,1,1,1,1,2] => 20 [2,1,1,1,2,1] => 21 [2,1,1,1,3] => 14 [2,1,1,2,1,1] => 22 [2,1,1,2,2] => 15 [2,1,1,3,1] => 16 [2,1,1,4] => 9 [2,1,2,1,1,1] => 23 [2,1,2,1,2] => 16 [2,1,2,2,1] => 17 [2,1,2,3] => 10 [2,1,3,1,1] => 18 [2,1,3,2] => 11 [2,1,4,1] => 12 [2,1,5] => 5 [2,2,1,1,1,1] => 24 [2,2,1,1,2] => 17 [2,2,1,2,1] => 18 [2,2,1,3] => 11 [2,2,2,1,1] => 19 [2,2,2,2] => 12 [2,2,3,1] => 13 [2,2,4] => 6 [2,3,1,1,1] => 20 [2,3,1,2] => 13 [2,3,2,1] => 14 [2,3,3] => 7 [2,4,1,1] => 15 [2,4,2] => 8 [2,5,1] => 9 [2,6] => 2 [3,1,1,1,1,1] => 25 [3,1,1,1,2] => 18 [3,1,1,2,1] => 19 [3,1,1,3] => 12 [3,1,2,1,1] => 20 [3,1,2,2] => 13 [3,1,3,1] => 14 [3,1,4] => 7 [3,2,1,1,1] => 21 [3,2,1,2] => 14 [3,2,2,1] => 15 [3,2,3] => 8 [3,3,1,1] => 16 [3,3,2] => 9 [3,4,1] => 10 [3,5] => 3 [4,1,1,1,1] => 22 [4,1,1,2] => 15 [4,1,2,1] => 16 [4,1,3] => 9 [4,2,1,1] => 17 [4,2,2] => 10 [4,3,1] => 11 [4,4] => 4 [5,1,1,1] => 18 [5,1,2] => 11 [5,2,1] => 12 [5,3] => 5 [6,1,1] => 13 [6,2] => 6 [7,1] => 7 [8] => 0 [1,1,1,1,4,1] => 18 [1,1,1,2,3,1] => 19 [1,1,2,1,3,1] => 20 [1,1,6,1] => 11 [1,2,1,1,3,1] => 21 [1,2,1,2,2,1] => 22 [1,2,2,1,2,1] => 23 [1,2,5,1] => 12 [1,3,1,1,2,1] => 24 [1,3,1,2,1,1] => 25 [1,3,2,1,1,1] => 26 [1,3,4,1] => 13 [1,4,1,1,1,1] => 27 [1,4,1,3] => 12 [1,4,2,2] => 13 [1,4,3,1] => 14 [1,5,1,2] => 14 [1,5,2,1] => 15 [1,6,1,1] => 16 [1,6,2] => 8 [1,7,1] => 9 [1,8] => 1 [2,1,5,1] => 13 [2,2,4,1] => 14 [3,1,4,1] => 15 [6,2,1] => 14 [7,1,1] => 15 [8,1] => 8 [1,1,1,1,1,1,1,1,1,1] => 45 [1,1,1,1,1,1,2,2] => 29 [1,1,1,1,2,1,1,2] => 31 [1,1,1,1,2,2,1,1] => 33 [1,1,1,1,3,3] => 17 [1,1,1,1,5,1] => 19 [1,1,2,1,1,1,1,2] => 33 [1,1,2,1,1,2,1,1] => 35 [1,1,2,1,2,3] => 19 [1,1,2,2,1,1,1,1] => 37 [1,1,2,2,2,2] => 21 [1,1,3,1,1,3] => 21 [1,1,3,2,1,2] => 23 [1,1,3,3,1,1] => 25 [1,1,4,4] => 9 [1,1,7,1] => 12 [1,2,1,1,4,1] => 22 [1,2,6,1] => 13 [1,3,1,1,3,1] => 25 [1,3,5,1] => 14 [1,4,1,1,2,1] => 28 [1,4,4,1] => 15 [1,5,1,1,1,1] => 31 [1,5,3,1] => 16 [1,6,1,2] => 16 [1,6,2,1] => 17 [1,7,1,1] => 18 [1,8,1] => 10 [1,9] => 1 [2,1,1,1,1,1,1,2] => 35 [2,1,1,1,1,2,1,1] => 37 [2,1,1,1,2,3] => 21 [2,1,1,2,1,1,1,1] => 39 [2,1,1,2,2,2] => 23 [2,1,2,1,1,3] => 23 [2,1,2,2,1,2] => 25 [2,1,2,3,1,1] => 27 [2,1,3,4] => 11 [2,1,6,1] => 14 [2,2,1,1,1,1,1,1] => 41 [2,2,1,1,2,2] => 25 [2,2,2,1,1,2] => 27 [2,2,2,2,1,1] => 29 [2,2,3,3] => 13 [3,1,1,1,1,3] => 25 [3,1,1,2,1,2] => 27 [3,1,1,3,1,1] => 29 [3,1,2,4] => 13 [3,2,1,1,1,2] => 29 [3,2,1,2,1,1] => 31 [3,2,2,3] => 15 [3,3,1,1,1,1] => 33 [3,3,2,2] => 17 [4,1,1,4] => 15 [4,2,1,3] => 17 [4,3,1,2] => 19 [4,4,1,1] => 21 [5,5] => 5 [8,1,1] => 17 [9,1] => 9 [1,10] => 1 [1,8,1,1] => 20 [1,7,2,1] => 19 [1,6,3,1] => 18 [1,5,4,1] => 17 [1,4,5,1] => 16 [1,3,6,1] => 15 [1,2,7,1] => 14 [1,1,8,1] => 13 [10,1] => 10 [1,1,1,1,1,1,1,1,1,1,1,1] => 66 [1,1,1,1,1,1,1,1,2,2] => 46 [1,1,1,1,1,1,2,2,1,1] => 50 [1,1,1,1,1,1,2,1,1,2] => 48 [1,1,1,1,1,1,3,3] => 30 [1,1,1,1,2,2,1,1,1,1] => 54 [1,1,1,1,2,2,2,2] => 34 [1,1,1,1,2,1,1,2,1,1] => 52 [1,1,1,1,2,1,1,1,1,2] => 50 [1,1,1,1,2,1,2,3] => 32 [1,1,1,1,3,3,1,1] => 38 [1,1,1,1,3,2,1,2] => 36 [1,1,1,1,3,1,1,3] => 34 [1,1,1,1,4,4] => 18 [1,1,2,2,1,1,1,1,1,1] => 58 [1,1,2,2,1,1,2,2] => 38 [1,1,2,2,2,2,1,1] => 42 [1,1,2,2,2,1,1,2] => 40 [1,1,2,2,3,3] => 22 [1,1,2,1,1,2,1,1,1,1] => 56 [1,1,2,1,1,2,2,2] => 36 [1,1,2,1,1,1,1,2,1,1] => 54 [1,1,2,1,1,1,1,1,1,2] => 52 [1,1,2,1,1,1,2,3] => 34 [1,1,2,1,2,3,1,1] => 40 [1,1,2,1,2,2,1,2] => 38 [1,1,2,1,2,1,1,3] => 36 [1,1,2,1,3,4] => 20 [1,1,3,3,1,1,1,1] => 46 [1,1,3,3,2,2] => 26 [1,1,3,2,1,2,1,1] => 44 [1,1,3,2,1,1,1,2] => 42 [1,1,3,2,2,3] => 24 [1,1,3,1,1,3,1,1] => 42 [1,1,3,1,1,2,1,2] => 40 [1,1,3,1,1,1,1,3] => 38 [1,1,3,1,2,4] => 22 [1,1,4,4,1,1] => 30 [1,1,4,3,1,2] => 28 [1,1,4,2,1,3] => 26 [1,1,4,1,1,4] => 24 [1,1,5,5] => 10 [2,2,1,1,1,1,1,1,1,1] => 62 [2,2,1,1,1,1,2,2] => 42 [2,2,1,1,2,2,1,1] => 46 [2,2,1,1,2,1,1,2] => 44 [2,2,1,1,3,3] => 26 [2,2,2,2,1,1,1,1] => 50 [2,2,2,2,2,2] => 30 [2,2,2,1,1,2,1,1] => 48 [2,2,2,1,1,1,1,2] => 46 [2,2,2,1,2,3] => 28 [2,2,3,3,1,1] => 34 [2,2,3,2,1,2] => 32 [2,2,3,1,1,3] => 30 [2,2,4,4] => 14 [2,1,1,2,1,1,1,1,1,1] => 60 [2,1,1,2,1,1,2,2] => 40 [2,1,1,2,2,2,1,1] => 44 [2,1,1,2,2,1,1,2] => 42 [2,1,1,2,3,3] => 24 [2,1,1,1,1,2,1,1,1,1] => 58 [2,1,1,1,1,2,2,2] => 38 [2,1,1,1,1,1,1,2,1,1] => 56 [2,1,1,1,1,1,1,1,1,2] => 54 [2,1,1,1,1,1,2,3] => 36 [2,1,1,1,2,3,1,1] => 42 [2,1,1,1,2,2,1,2] => 40 [2,1,1,1,2,1,1,3] => 38 [2,1,1,1,3,4] => 22 [2,1,2,3,1,1,1,1] => 48 [2,1,2,3,2,2] => 28 [2,1,2,2,1,2,1,1] => 46 [2,1,2,2,1,1,1,2] => 44 [2,1,2,2,2,3] => 26 [2,1,2,1,1,3,1,1] => 44 [2,1,2,1,1,2,1,2] => 42 [2,1,2,1,1,1,1,3] => 40 [2,1,2,1,2,4] => 24 [2,1,3,4,1,1] => 32 [2,1,3,3,1,2] => 30 [2,1,3,2,1,3] => 28 [2,1,3,1,1,4] => 26 [2,1,4,5] => 12 [3,3,1,1,1,1,1,1] => 54 [3,3,1,1,2,2] => 34 [3,3,2,2,1,1] => 38 [3,3,2,1,1,2] => 36 [3,3,3,3] => 18 [3,2,1,2,1,1,1,1] => 52 [3,2,1,2,2,2] => 32 [3,2,1,1,1,2,1,1] => 50 [3,2,1,1,1,1,1,2] => 48 [3,2,1,1,2,3] => 30 [3,2,2,3,1,1] => 36 [3,2,2,2,1,2] => 34 [3,2,2,1,1,3] => 32 [3,2,3,4] => 16 [3,1,1,3,1,1,1,1] => 50 [3,1,1,3,2,2] => 30 [3,1,1,2,1,2,1,1] => 48 [3,1,1,2,1,1,1,2] => 46 [3,1,1,2,2,3] => 28 [3,1,1,1,1,3,1,1] => 46 [3,1,1,1,1,2,1,2] => 44 [3,1,1,1,1,1,1,3] => 42 [3,1,1,1,2,4] => 26 [3,1,2,4,1,1] => 34 [3,1,2,3,1,2] => 32 [3,1,2,2,1,3] => 30 [3,1,2,1,1,4] => 28 [3,1,3,5] => 14 [4,4,1,1,1,1] => 42 [4,4,2,2] => 22 [4,3,1,2,1,1] => 40 [4,3,1,1,1,2] => 38 [4,3,2,3] => 20 [4,2,1,3,1,1] => 38 [4,2,1,2,1,2] => 36 [4,2,1,1,1,3] => 34 [4,2,2,4] => 18 [4,1,1,4,1,1] => 36 [4,1,1,3,1,2] => 34 [4,1,1,2,1,3] => 32 [4,1,1,1,1,4] => 30 [4,1,2,5] => 16 [5,5,1,1] => 26 [5,4,1,2] => 24 [5,3,1,3] => 22 [5,2,1,4] => 20 [5,1,1,5] => 18 [6,6] => 6 ----------------------------------------------------------------------------- Created: Sep 21, 2011 at 16:41 by Chris Berg ----------------------------------------------------------------------------- Last Updated: Jan 26, 2018 at 09:17 by Martin Rubey