***************************************************************************** * 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: St000033 ----------------------------------------------------------------------------- Collection: Permutations ----------------------------------------------------------------------------- Description: The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. ----------------------------------------------------------------------------- References: [[http://en.wikipedia.org/wiki/Bruhat_order]] ----------------------------------------------------------------------------- Code: def statistic(x): return x.bruhat_greater().cardinality() ----------------------------------------------------------------------------- Statistic values: [1] => 1 [1,2] => 2 [2,1] => 1 [1,2,3] => 6 [1,3,2] => 4 [2,1,3] => 4 [2,3,1] => 2 [3,1,2] => 2 [3,2,1] => 1 [1,2,3,4] => 24 [1,2,4,3] => 18 [1,3,2,4] => 20 [1,3,4,2] => 12 [1,4,2,3] => 12 [1,4,3,2] => 8 [2,1,3,4] => 18 [2,1,4,3] => 14 [2,3,1,4] => 12 [2,3,4,1] => 6 [2,4,1,3] => 8 [2,4,3,1] => 4 [3,1,2,4] => 12 [3,1,4,2] => 8 [3,2,1,4] => 8 [3,2,4,1] => 4 [3,4,1,2] => 4 [3,4,2,1] => 2 [4,1,2,3] => 6 [4,1,3,2] => 4 [4,2,1,3] => 4 [4,2,3,1] => 2 [4,3,1,2] => 2 [4,3,2,1] => 1 [1,2,3,4,5] => 120 [1,2,3,5,4] => 96 [1,2,4,3,5] => 108 [1,2,4,5,3] => 72 [1,2,5,3,4] => 72 [1,2,5,4,3] => 54 [1,3,2,4,5] => 108 [1,3,2,5,4] => 88 [1,3,4,2,5] => 84 [1,3,4,5,2] => 48 [1,3,5,2,4] => 60 [1,3,5,4,2] => 36 [1,4,2,3,5] => 84 [1,4,2,5,3] => 60 [1,4,3,2,5] => 68 [1,4,3,5,2] => 40 [1,4,5,2,3] => 36 [1,4,5,3,2] => 24 [1,5,2,3,4] => 48 [1,5,2,4,3] => 36 [1,5,3,2,4] => 40 [1,5,3,4,2] => 24 [1,5,4,2,3] => 24 [1,5,4,3,2] => 16 [2,1,3,4,5] => 96 [2,1,3,5,4] => 78 [2,1,4,3,5] => 88 [2,1,4,5,3] => 60 [2,1,5,3,4] => 60 [2,1,5,4,3] => 46 [2,3,1,4,5] => 72 [2,3,1,5,4] => 60 [2,3,4,1,5] => 48 [2,3,4,5,1] => 24 [2,3,5,1,4] => 36 [2,3,5,4,1] => 18 [2,4,1,3,5] => 60 [2,4,1,5,3] => 44 [2,4,3,1,5] => 40 [2,4,3,5,1] => 20 [2,4,5,1,3] => 24 [2,4,5,3,1] => 12 [2,5,1,3,4] => 36 [2,5,1,4,3] => 28 [2,5,3,1,4] => 24 [2,5,3,4,1] => 12 [2,5,4,1,3] => 16 [2,5,4,3,1] => 8 [3,1,2,4,5] => 72 [3,1,2,5,4] => 60 [3,1,4,2,5] => 60 [3,1,4,5,2] => 36 [3,1,5,2,4] => 44 [3,1,5,4,2] => 28 [3,2,1,4,5] => 54 [3,2,1,5,4] => 46 [3,2,4,1,5] => 36 [3,2,4,5,1] => 18 [3,2,5,1,4] => 28 [3,2,5,4,1] => 14 [3,4,1,2,5] => 36 [3,4,1,5,2] => 24 [3,4,2,1,5] => 24 [3,4,2,5,1] => 12 [3,4,5,1,2] => 12 [3,4,5,2,1] => 6 [3,5,1,2,4] => 24 [3,5,1,4,2] => 16 [3,5,2,1,4] => 16 [3,5,2,4,1] => 8 [3,5,4,1,2] => 8 [3,5,4,2,1] => 4 [4,1,2,3,5] => 48 [4,1,2,5,3] => 36 [4,1,3,2,5] => 40 [4,1,3,5,2] => 24 [4,1,5,2,3] => 24 [4,1,5,3,2] => 16 [4,2,1,3,5] => 36 [4,2,1,5,3] => 28 [4,2,3,1,5] => 24 [4,2,3,5,1] => 12 [4,2,5,1,3] => 16 [4,2,5,3,1] => 8 [4,3,1,2,5] => 24 [4,3,1,5,2] => 16 [4,3,2,1,5] => 16 [4,3,2,5,1] => 8 [4,3,5,1,2] => 8 [4,3,5,2,1] => 4 [4,5,1,2,3] => 12 [4,5,1,3,2] => 8 [4,5,2,1,3] => 8 [4,5,2,3,1] => 4 [4,5,3,1,2] => 4 [4,5,3,2,1] => 2 [5,1,2,3,4] => 24 [5,1,2,4,3] => 18 [5,1,3,2,4] => 20 [5,1,3,4,2] => 12 [5,1,4,2,3] => 12 [5,1,4,3,2] => 8 [5,2,1,3,4] => 18 [5,2,1,4,3] => 14 [5,2,3,1,4] => 12 [5,2,3,4,1] => 6 [5,2,4,1,3] => 8 [5,2,4,3,1] => 4 [5,3,1,2,4] => 12 [5,3,1,4,2] => 8 [5,3,2,1,4] => 8 [5,3,2,4,1] => 4 [5,3,4,1,2] => 4 [5,3,4,2,1] => 2 [5,4,1,2,3] => 6 [5,4,1,3,2] => 4 [5,4,2,1,3] => 4 [5,4,2,3,1] => 2 [5,4,3,1,2] => 2 [5,4,3,2,1] => 1 ----------------------------------------------------------------------------- Created: Feb 13, 2013 at 01:29 by Sara Billey ----------------------------------------------------------------------------- Last Updated: Sep 13, 2014 at 21:22 by Martin Rubey