***************************************************************************** * 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: St000958 ----------------------------------------------------------------------------- Collection: Permutations ----------------------------------------------------------------------------- Description: The number of Bruhat factorizations of a permutation. This is the number of factorizations $\pi = t_1 \cdots t_\ell$ for transpositions $\{ t_i \mid 1 \leq i \leq \ell\}$ such that the number of inversions of $t_1 \cdots t_i$ equals $i$ for all $1 \leq i \leq \ell$. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: @cached_function def bruhat_poset(n): return Permutations(n).bruhat_poset(facade=True) def statistic(pi): P = bruhat_poset(len(pi)) I = P.subposet(P.principal_order_ideal(pi)) return len(I.maximal_chains()) ----------------------------------------------------------------------------- Statistic values: [1] => 1 [1,2] => 1 [2,1] => 1 [1,2,3] => 1 [1,3,2] => 1 [2,1,3] => 1 [2,3,1] => 2 [3,1,2] => 2 [3,2,1] => 4 [1,2,3,4] => 1 [1,2,4,3] => 1 [1,3,2,4] => 1 [1,3,4,2] => 2 [1,4,2,3] => 2 [1,4,3,2] => 4 [2,1,3,4] => 1 [2,1,4,3] => 2 [2,3,1,4] => 2 [2,3,4,1] => 6 [2,4,1,3] => 6 [2,4,3,1] => 16 [3,1,2,4] => 2 [3,1,4,2] => 6 [3,2,1,4] => 4 [3,2,4,1] => 16 [3,4,1,2] => 20 [3,4,2,1] => 52 [4,1,2,3] => 6 [4,1,3,2] => 16 [4,2,1,3] => 16 [4,2,3,1] => 64 [4,3,1,2] => 52 [4,3,2,1] => 168 [1,2,3,4,5] => 1 [1,2,3,5,4] => 1 [1,2,4,3,5] => 1 [1,2,4,5,3] => 2 [1,2,5,3,4] => 2 [1,2,5,4,3] => 4 [1,3,2,4,5] => 1 [1,3,2,5,4] => 2 [1,3,4,2,5] => 2 [1,3,4,5,2] => 6 [1,3,5,2,4] => 6 [1,3,5,4,2] => 16 [1,4,2,3,5] => 2 [1,4,2,5,3] => 6 [1,4,3,2,5] => 4 [1,4,3,5,2] => 16 [1,4,5,2,3] => 20 [1,4,5,3,2] => 52 [1,5,2,3,4] => 6 [1,5,2,4,3] => 16 [1,5,3,2,4] => 16 [1,5,3,4,2] => 64 [1,5,4,2,3] => 52 [1,5,4,3,2] => 168 [2,1,3,4,5] => 1 [2,1,3,5,4] => 2 [2,1,4,3,5] => 2 [2,1,4,5,3] => 6 [2,1,5,3,4] => 6 [2,1,5,4,3] => 16 [2,3,1,4,5] => 2 [2,3,1,5,4] => 6 [2,3,4,1,5] => 6 [2,3,4,5,1] => 24 [2,3,5,1,4] => 24 [2,3,5,4,1] => 80 [2,4,1,3,5] => 6 [2,4,1,5,3] => 24 [2,4,3,1,5] => 16 [2,4,3,5,1] => 80 [2,4,5,1,3] => 100 [2,4,5,3,1] => 312 [2,5,1,3,4] => 24 [2,5,1,4,3] => 80 [2,5,3,1,4] => 80 [2,5,3,4,1] => 384 [2,5,4,1,3] => 312 [2,5,4,3,1] => 1176 [3,1,2,4,5] => 2 [3,1,2,5,4] => 6 [3,1,4,2,5] => 6 [3,1,4,5,2] => 24 [3,1,5,2,4] => 24 [3,1,5,4,2] => 80 [3,2,1,4,5] => 4 [3,2,1,5,4] => 16 [3,2,4,1,5] => 16 [3,2,4,5,1] => 80 [3,2,5,1,4] => 80 [3,2,5,4,1] => 320 [3,4,1,2,5] => 20 [3,4,1,5,2] => 100 [3,4,2,1,5] => 52 [3,4,2,5,1] => 312 [3,4,5,1,2] => 464 [3,4,5,2,1] => 1408 [3,5,1,2,4] => 100 [3,5,1,4,2] => 424 [3,5,2,1,4] => 312 [3,5,2,4,1] => 1752 [3,5,4,1,2] => 1680 [3,5,4,2,1] => 6016 [4,1,2,3,5] => 6 [4,1,2,5,3] => 24 [4,1,3,2,5] => 16 [4,1,3,5,2] => 80 [4,1,5,2,3] => 100 [4,1,5,3,2] => 312 [4,2,1,3,5] => 16 [4,2,1,5,3] => 80 [4,2,3,1,5] => 64 [4,2,3,5,1] => 384 [4,2,5,1,3] => 424 [4,2,5,3,1] => 1752 [4,3,1,2,5] => 52 [4,3,1,5,2] => 312 [4,3,2,1,5] => 168 [4,3,2,5,1] => 1176 [4,3,5,1,2] => 1680 [4,3,5,2,1] => 6016 [4,5,1,2,3] => 464 [4,5,1,3,2] => 1680 [4,5,2,1,3] => 1680 [4,5,2,3,1] => 9216 [4,5,3,1,2] => 6720 [4,5,3,2,1] => 27968 [5,1,2,3,4] => 24 [5,1,2,4,3] => 80 [5,1,3,2,4] => 80 [5,1,3,4,2] => 384 [5,1,4,2,3] => 312 [5,1,4,3,2] => 1176 [5,2,1,3,4] => 80 [5,2,1,4,3] => 320 [5,2,3,1,4] => 384 [5,2,3,4,1] => 2176 [5,2,4,1,3] => 1752 [5,2,4,3,1] => 8032 [5,3,1,2,4] => 312 [5,3,1,4,2] => 1752 [5,3,2,1,4] => 1176 [5,3,2,4,1] => 8032 [5,3,4,1,2] => 9216 [5,3,4,2,1] => 37312 [5,4,1,2,3] => 1408 [5,4,1,3,2] => 6016 [5,4,2,1,3] => 6016 [5,4,2,3,1] => 37312 [5,4,3,1,2] => 27968 [5,4,3,2,1] => 130560 ----------------------------------------------------------------------------- Created: Aug 28, 2017 at 11:08 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Jan 13, 2018 at 12:45 by Martin Rubey