***************************************************************************** * 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: St001768 ----------------------------------------------------------------------------- Collection: Signed permutations ----------------------------------------------------------------------------- Description: The number of reduced words of a signed permutation. This is the number of ways to write a permutation as a minimal length product of simple reflections. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(pi): return len(pi.reduced_words()) ----------------------------------------------------------------------------- Statistic values: [1] => 1 [-1] => 1 [1,2] => 1 [1,-2] => 1 [-1,2] => 1 [-1,-2] => 2 [2,1] => 1 [2,-1] => 1 [-2,1] => 1 [-2,-1] => 1 [1,2,3] => 1 [1,2,-3] => 1 [1,-2,3] => 1 [1,-2,-3] => 2 [-1,2,3] => 1 [-1,2,-3] => 4 [-1,-2,3] => 14 [-1,-2,-3] => 42 [1,3,2] => 1 [1,3,-2] => 1 [1,-3,2] => 1 [1,-3,-2] => 1 [-1,3,2] => 4 [-1,3,-2] => 9 [-1,-3,2] => 9 [-1,-3,-2] => 16 [2,1,3] => 1 [2,1,-3] => 2 [2,-1,3] => 1 [2,-1,-3] => 3 [-2,1,3] => 1 [-2,1,-3] => 3 [-2,-1,3] => 5 [-2,-1,-3] => 12 [2,3,1] => 1 [2,3,-1] => 1 [2,-3,1] => 2 [2,-3,-1] => 2 [-2,3,1] => 3 [-2,3,-1] => 5 [-2,-3,1] => 5 [-2,-3,-1] => 7 [3,1,2] => 1 [3,1,-2] => 2 [3,-1,2] => 3 [3,-1,-2] => 5 [-3,1,2] => 1 [-3,1,-2] => 2 [-3,-1,2] => 5 [-3,-1,-2] => 7 [3,2,1] => 2 [3,2,-1] => 3 [3,-2,1] => 2 [3,-2,-1] => 2 [-3,2,1] => 3 [-3,2,-1] => 5 [-3,-2,1] => 2 [-3,-2,-1] => 2 [1,2,3,4] => 1 [1,2,3,-4] => 1 [1,2,-3,4] => 1 [1,2,-3,-4] => 2 [1,-2,3,4] => 1 [1,-2,3,-4] => 4 [1,-2,-3,4] => 14 [1,-2,-3,-4] => 42 [-1,2,3,4] => 1 [-1,2,3,-4] => 6 [-1,2,-3,4] => 48 [-1,2,-3,-4] => 198 [-1,-2,3,4] => 132 [-1,-2,3,-4] => 858 [-1,-2,-3,4] => 6006 [-1,-2,-3,-4] => 24024 [1,2,4,3] => 1 [1,2,4,-3] => 1 [1,2,-4,3] => 1 [1,2,-4,-3] => 1 [1,-2,4,3] => 4 [1,-2,4,-3] => 9 [1,-2,-4,3] => 9 [1,-2,-4,-3] => 16 [-1,2,4,3] => 6 [-1,2,4,-3] => 20 [-1,2,-4,3] => 20 [-1,2,-4,-3] => 50 [-1,-2,4,3] => 858 [-1,-2,4,-3] => 2860 [-1,-2,-4,3] => 2860 [-1,-2,-4,-3] => 7150 [1,3,2,4] => 1 [1,3,2,-4] => 2 [1,3,-2,4] => 1 [1,3,-2,-4] => 3 [1,-3,2,4] => 1 [1,-3,2,-4] => 3 [1,-3,-2,4] => 5 [1,-3,-2,-4] => 12 [-1,3,2,4] => 6 [-1,3,2,-4] => 40 [-1,3,-2,4] => 90 [-1,3,-2,-4] => 486 [-1,-3,2,4] => 90 [-1,-3,2,-4] => 486 [-1,-3,-2,4] => 1872 [-1,-3,-2,-4] => 6318 [1,3,4,2] => 1 [1,3,4,-2] => 1 [1,3,-4,2] => 2 [1,3,-4,-2] => 2 [1,-3,4,2] => 3 [1,-3,4,-2] => 5 [1,-3,-4,2] => 5 [1,-3,-4,-2] => 7 [-1,3,4,2] => 20 [-1,3,4,-2] => 48 [-1,3,-4,2] => 98 [-1,3,-4,-2] => 198 [-1,-3,4,2] => 486 [-1,-3,4,-2] => 1300 [-1,-3,-4,2] => 1300 [-1,-3,-4,-2] => 2730 [1,4,2,3] => 1 [1,4,2,-3] => 2 [1,4,-2,3] => 3 [1,4,-2,-3] => 5 [1,-4,2,3] => 1 [1,-4,2,-3] => 2 [1,-4,-2,3] => 5 [1,-4,-2,-3] => 7 [-1,4,2,3] => 20 [-1,4,2,-3] => 98 [-1,4,-2,3] => 486 [-1,4,-2,-3] => 1300 [-1,-4,2,3] => 48 [-1,-4,2,-3] => 198 [-1,-4,-2,3] => 1300 [-1,-4,-2,-3] => 2730 [1,4,3,2] => 2 [1,4,3,-2] => 3 [1,4,-3,2] => 2 [1,4,-3,-2] => 2 [1,-4,3,2] => 3 [1,-4,3,-2] => 5 [1,-4,-3,2] => 2 [1,-4,-3,-2] => 2 [-1,4,3,2] => 98 [-1,4,3,-2] => 288 [-1,4,-3,2] => 198 [-1,4,-3,-2] => 352 [-1,-4,3,2] => 288 [-1,-4,3,-2] => 838 [-1,-4,-3,2] => 352 [-1,-4,-3,-2] => 572 [2,1,3,4] => 1 [2,1,3,-4] => 2 [2,1,-3,4] => 4 [2,1,-3,-4] => 10 [2,-1,3,4] => 1 [2,-1,3,-4] => 5 [2,-1,-3,4] => 28 [2,-1,-3,-4] => 100 [-2,1,3,4] => 1 [-2,1,3,-4] => 5 [-2,1,-3,4] => 28 [-2,1,-3,-4] => 100 [-2,-1,3,4] => 42 [-2,-1,3,-4] => 240 [-2,-1,-3,4] => 1274 [-2,-1,-3,-4] => 4550 [2,1,4,3] => 2 [2,1,4,-3] => 3 [2,1,-4,3] => 3 [2,1,-4,-3] => 4 [2,-1,4,3] => 5 [2,-1,4,-3] => 14 [2,-1,-4,3] => 14 [2,-1,-4,-3] => 30 [-2,1,4,3] => 5 [-2,1,4,-3] => 14 [-2,1,-4,3] => 14 [-2,1,-4,-3] => 30 [-2,-1,4,3] => 240 [-2,-1,4,-3] => 702 [-2,-1,-4,3] => 702 [-2,-1,-4,-3] => 1560 [2,3,1,4] => 1 [2,3,1,-4] => 3 [2,3,-1,4] => 1 [2,3,-1,-4] => 4 [2,-3,1,4] => 4 [2,-3,1,-4] => 14 [2,-3,-1,4] => 14 [2,-3,-1,-4] => 42 [-2,3,1,4] => 5 [-2,3,1,-4] => 28 [-2,3,-1,4] => 42 [-2,3,-1,-4] => 198 [-2,-3,1,4] => 42 [-2,-3,1,-4] => 198 [-2,-3,-1,4] => 572 [-2,-3,-1,-4] => 1716 [2,3,4,1] => 1 [2,3,4,-1] => 1 [2,3,-4,1] => 3 [2,3,-4,-1] => 3 [2,-3,4,1] => 9 [2,-3,4,-1] => 14 [2,-3,-4,1] => 21 [2,-3,-4,-1] => 28 [-2,3,4,1] => 14 [-2,3,4,-1] => 28 [-2,3,-4,1] => 58 [-2,3,-4,-1] => 100 [-2,-3,4,1] => 198 [-2,-3,4,-1] => 462 [-2,-3,-4,1] => 462 [-2,-3,-4,-1] => 858 [2,4,1,3] => 2 [2,4,1,-3] => 5 [2,4,-1,3] => 4 [2,4,-1,-3] => 9 [2,-4,1,3] => 3 [2,-4,1,-3] => 7 [2,-4,-1,3] => 9 [2,-4,-1,-3] => 16 [-2,4,1,3] => 14 [-2,4,1,-3] => 58 [-2,4,-1,3] => 198 [-2,4,-1,-3] => 462 [-2,-4,1,3] => 28 [-2,-4,1,-3] => 100 [-2,-4,-1,3] => 462 [-2,-4,-1,-3] => 858 [2,4,3,1] => 3 [2,4,3,-1] => 4 [2,4,-3,1] => 5 [2,4,-3,-1] => 5 [2,-4,3,1] => 6 [2,-4,3,-1] => 9 [2,-4,-3,1] => 7 [2,-4,-3,-1] => 7 [-2,4,3,1] => 58 [-2,4,3,-1] => 142 [-2,4,-3,1] => 100 [-2,4,-3,-1] => 154 [-2,-4,3,1] => 142 [-2,-4,3,-1] => 352 [-2,-4,-3,1] => 154 [-2,-4,-3,-1] => 220 [3,1,2,4] => 1 [3,1,2,-4] => 3 [3,1,-2,4] => 4 [3,1,-2,-4] => 14 [3,-1,2,4] => 5 [3,-1,2,-4] => 28 [3,-1,-2,4] => 42 [3,-1,-2,-4] => 198 [-3,1,2,4] => 1 [-3,1,2,-4] => 4 [-3,1,-2,4] => 14 [-3,1,-2,-4] => 42 [-3,-1,2,4] => 42 [-3,-1,2,-4] => 198 [-3,-1,-2,4] => 572 [-3,-1,-2,-4] => 1716 [3,1,4,2] => 2 [3,1,4,-2] => 3 [3,1,-4,2] => 5 [3,1,-4,-2] => 7 [3,-1,4,2] => 14 [3,-1,4,-2] => 28 [3,-1,-4,2] => 58 [3,-1,-4,-2] => 100 [-3,1,4,2] => 4 [-3,1,4,-2] => 9 [-3,1,-4,2] => 9 [-3,1,-4,-2] => 16 [-3,-1,4,2] => 198 [-3,-1,4,-2] => 462 [-3,-1,-4,2] => 462 [-3,-1,-4,-2] => 858 [3,2,1,4] => 2 [3,2,1,-4] => 8 [3,2,-1,4] => 5 [3,2,-1,-4] => 23 [3,-2,1,4] => 4 [3,-2,1,-4] => 18 [3,-2,-1,4] => 14 [3,-2,-1,-4] => 56 [-3,2,1,4] => 5 [-3,2,1,-4] => 23 [-3,2,-1,4] => 42 [-3,2,-1,-4] => 156 [-3,-2,1,4] => 14 [-3,-2,1,-4] => 56 [-3,-2,-1,4] => 110 [-3,-2,-1,-4] => 286 [3,2,4,1] => 3 [3,2,4,-1] => 4 [3,2,-4,1] => 11 [3,2,-4,-1] => 14 [3,-2,4,1] => 9 [3,-2,4,-1] => 14 [3,-2,-4,1] => 30 [3,-2,-4,-1] => 42 [-3,2,4,1] => 14 [-3,2,4,-1] => 28 [-3,2,-4,1] => 44 [-3,2,-4,-1] => 72 [-3,-2,4,1] => 56 [-3,-2,4,-1] => 110 [-3,-2,-4,1] => 110 [-3,-2,-4,-1] => 176 [3,4,1,2] => 2 [3,4,1,-2] => 5 [3,4,-1,2] => 9 [3,4,-1,-2] => 14 [3,-4,1,2] => 5 [3,-4,1,-2] => 12 [3,-4,-1,2] => 30 [3,-4,-1,-2] => 42 [-3,4,1,2] => 9 [-3,4,1,-2] => 30 [-3,4,-1,2] => 156 [-3,4,-1,-2] => 264 [-3,-4,1,2] => 14 [-3,-4,1,-2] => 42 [-3,-4,-1,2] => 264 [-3,-4,-1,-2] => 396 [3,4,2,1] => 5 [3,4,2,-1] => 9 [3,4,-2,1] => 5 [3,4,-2,-1] => 5 [3,-4,2,1] => 16 [3,-4,2,-1] => 30 [3,-4,-2,1] => 12 [3,-4,-2,-1] => 12 [-3,4,2,1] => 44 [-3,4,2,-1] => 114 [-3,4,-2,1] => 42 [-3,4,-2,-1] => 54 [-3,-4,2,1] => 84 [-3,-4,2,-1] => 210 [-3,-4,-2,1] => 54 [-3,-4,-2,-1] => 66 [4,1,2,3] => 1 [4,1,2,-3] => 3 [4,1,-2,3] => 9 [4,1,-2,-3] => 21 [4,-1,2,3] => 14 [4,-1,2,-3] => 58 [4,-1,-2,3] => 198 [4,-1,-2,-3] => 462 [-4,1,2,3] => 1 [-4,1,2,-3] => 3 [-4,1,-2,3] => 14 [-4,1,-2,-3] => 28 [-4,-1,2,3] => 28 [-4,-1,2,-3] => 100 [-4,-1,-2,3] => 462 [-4,-1,-2,-3] => 858 [4,1,3,2] => 3 [4,1,3,-2] => 6 [4,1,-3,2] => 5 [4,1,-3,-2] => 7 [4,-1,3,2] => 58 [4,-1,3,-2] => 142 [4,-1,-3,2] => 100 [4,-1,-3,-2] => 154 [-4,1,3,2] => 4 [-4,1,3,-2] => 9 [-4,1,-3,2] => 5 [-4,1,-3,-2] => 7 [-4,-1,3,2] => 142 [-4,-1,3,-2] => 352 [-4,-1,-3,2] => 154 [-4,-1,-3,-2] => 220 [4,2,1,3] => 3 [4,2,1,-3] => 11 [4,2,-1,3] => 14 [4,2,-1,-3] => 44 [4,-2,1,3] => 9 [4,-2,1,-3] => 30 [4,-2,-1,3] => 56 [4,-2,-1,-3] => 110 [-4,2,1,3] => 4 [-4,2,1,-3] => 14 [-4,2,-1,3] => 28 [-4,2,-1,-3] => 72 [-4,-2,1,3] => 14 [-4,-2,1,-3] => 42 [-4,-2,-1,3] => 110 [-4,-2,-1,-3] => 176 [4,2,3,1] => 6 [4,2,3,-1] => 10 [4,2,-3,1] => 16 [4,2,-3,-1] => 21 [4,-2,3,1] => 30 [4,-2,3,-1] => 56 [4,-2,-3,1] => 42 [4,-2,-3,-1] => 54 [-4,2,3,1] => 10 [-4,2,3,-1] => 19 [-4,2,-3,1] => 21 [-4,2,-3,-1] => 28 [-4,-2,3,1] => 56 [-4,-2,3,-1] => 110 [-4,-2,-3,1] => 54 [-4,-2,-3,-1] => 66 [4,3,1,2] => 5 [4,3,1,-2] => 16 [4,3,-1,2] => 44 [4,3,-1,-2] => 84 [4,-3,1,2] => 5 [4,-3,1,-2] => 12 [4,-3,-1,2] => 42 [4,-3,-1,-2] => 54 [-4,3,1,2] => 9 [-4,3,1,-2] => 30 [-4,3,-1,2] => 114 [-4,3,-1,-2] => 210 [-4,-3,1,2] => 5 [-4,-3,1,-2] => 12 [-4,-3,-1,2] => 54 [-4,-3,-1,-2] => 66 [4,3,2,1] => 16 [4,3,2,-1] => 35 [4,3,-2,1] => 21 [4,3,-2,-1] => 26 [4,-3,2,1] => 21 [4,-3,2,-1] => 42 [4,-3,-2,1] => 12 [4,-3,-2,-1] => 12 [-4,3,2,1] => 35 [-4,3,2,-1] => 84 [-4,3,-2,1] => 42 [-4,3,-2,-1] => 54 [-4,-3,2,1] => 26 [-4,-3,2,-1] => 54 [-4,-3,-2,1] => 12 [-4,-3,-2,-1] => 12 ----------------------------------------------------------------------------- Created: Feb 04, 2022 at 13:13 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Feb 04, 2022 at 13:13 by Martin Rubey