***************************************************************************** * 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: St001822 ----------------------------------------------------------------------------- Collection: Signed permutations ----------------------------------------------------------------------------- Description: The number of alignments of a signed permutation. An alignment of a signed permutation $n\in\mathfrak H_n$ is either a nesting alignment, [[St001866]], an alignment of type EN, [[St001867]], or an alignment of type NE, [[St001868]]. Let $\operatorname{al}$ be the number of alignments of $\pi$, let \operatorname{cr} be the number of crossings, [[St001862]], let \operatorname{wex} be the number of weak excedances, [[St001863]], and let \operatorname{neg} be the number of negative entries, [[St001429]]. Then, $\operatorname{al}+\operatorname{cr}=(n-\operatorname{wex})(\operatorname{wex}-1+\operatorname{neg})+\binom{\operatorname{neg}{2}$. ----------------------------------------------------------------------------- References: [1] Cho, S., Park, K. Permutation statistics and weak Bruhat order in permutation tableaux of type $B$ [[MathSciNet:3319075]] ----------------------------------------------------------------------------- Code: def statistic(pi): n = len(pi) A_nest = [(i, j) for i in range(1, n+1) for j in range(1, n+1) if (-i < -j < -pi(j) < -pi(i)) or (-i < j <= pi(j) < -pi(i)) or (i < j <= pi(j) < pi(i))] A_EN = [(i, j) for i in range(1, n+1) for j in range(1, n+1) if (-i < 0 < -pi(i) < pi(j) < j) or (i <= pi(i) < pi(j) < j)] A_NE = [(i, j) for i in range(1, n+1) for j in range(1, n+1) if (pi(i) < i < j <= pi(j))] return len(A_nest + A_EN + A_NE) ----------------------------------------------------------------------------- Statistic values: [1] => 0 [-1] => 0 [1,2] => 0 [1,-2] => 1 [-1,2] => 1 [-1,-2] => 1 [2,1] => 0 [2,-1] => 0 [-2,1] => 0 [-2,-1] => 0 [1,2,3] => 0 [1,2,-3] => 2 [1,-2,3] => 2 [1,-2,-3] => 3 [-1,2,3] => 2 [-1,2,-3] => 3 [-1,-2,3] => 3 [-1,-2,-3] => 3 [1,3,2] => 1 [1,3,-2] => 1 [1,-3,2] => 2 [1,-3,-2] => 2 [-1,3,2] => 2 [-1,3,-2] => 2 [-1,-3,2] => 2 [-1,-3,-2] => 2 [2,1,3] => 1 [2,1,-3] => 2 [2,-1,3] => 1 [2,-1,-3] => 2 [-2,1,3] => 2 [-2,1,-3] => 2 [-2,-1,3] => 2 [-2,-1,-3] => 2 [2,3,1] => 0 [2,3,-1] => 0 [2,-3,1] => 1 [2,-3,-1] => 1 [-2,3,1] => 1 [-2,3,-1] => 1 [-2,-3,1] => 1 [-2,-3,-1] => 1 [3,1,2] => 0 [3,1,-2] => 1 [3,-1,2] => 1 [3,-1,-2] => 1 [-3,1,2] => 0 [-3,1,-2] => 1 [-3,-1,2] => 1 [-3,-1,-2] => 1 [3,2,1] => 1 [3,2,-1] => 1 [3,-2,1] => 0 [3,-2,-1] => 0 [-3,2,1] => 2 [-3,2,-1] => 2 [-3,-2,1] => 0 [-3,-2,-1] => 0 [1,2,3,4] => 0 [1,2,3,-4] => 3 [1,2,-3,4] => 3 [1,2,-3,-4] => 5 [1,-2,3,4] => 3 [1,-2,3,-4] => 5 [1,-2,-3,4] => 5 [1,-2,-3,-4] => 6 [-1,2,3,4] => 3 [-1,2,3,-4] => 5 [-1,2,-3,4] => 5 [-1,2,-3,-4] => 6 [-1,-2,3,4] => 5 [-1,-2,3,-4] => 6 [-1,-2,-3,4] => 6 [-1,-2,-3,-4] => 6 [1,2,4,3] => 2 [1,2,4,-3] => 2 [1,2,-4,3] => 4 [1,2,-4,-3] => 4 [1,-2,4,3] => 4 [1,-2,4,-3] => 4 [1,-2,-4,3] => 5 [1,-2,-4,-3] => 5 [-1,2,4,3] => 4 [-1,2,4,-3] => 4 [-1,2,-4,3] => 5 [-1,2,-4,-3] => 5 [-1,-2,4,3] => 5 [-1,-2,4,-3] => 5 [-1,-2,-4,3] => 5 [-1,-2,-4,-3] => 5 [1,3,2,4] => 2 [1,3,2,-4] => 4 [1,3,-2,4] => 2 [1,3,-2,-4] => 4 [1,-3,2,4] => 4 [1,-3,2,-4] => 5 [1,-3,-2,4] => 4 [1,-3,-2,-4] => 5 [-1,3,2,4] => 4 [-1,3,2,-4] => 5 [-1,3,-2,4] => 4 [-1,3,-2,-4] => 5 [-1,-3,2,4] => 5 [-1,-3,2,-4] => 5 [-1,-3,-2,4] => 5 [-1,-3,-2,-4] => 5 [1,3,4,2] => 1 [1,3,4,-2] => 1 [1,3,-4,2] => 3 [1,3,-4,-2] => 3 [1,-3,4,2] => 3 [1,-3,4,-2] => 3 [1,-3,-4,2] => 4 [1,-3,-4,-2] => 4 [-1,3,4,2] => 3 [-1,3,4,-2] => 3 [-1,3,-4,2] => 4 [-1,3,-4,-2] => 4 [-1,-3,4,2] => 4 [-1,-3,4,-2] => 4 [-1,-3,-4,2] => 4 [-1,-3,-4,-2] => 4 [1,4,2,3] => 2 [1,4,2,-3] => 3 [1,4,-2,3] => 3 [1,4,-2,-3] => 3 [1,-4,2,3] => 3 [1,-4,2,-3] => 4 [1,-4,-2,3] => 4 [1,-4,-2,-3] => 4 [-1,4,2,3] => 3 [-1,4,2,-3] => 4 [-1,4,-2,3] => 4 [-1,4,-2,-3] => 4 [-1,-4,2,3] => 3 [-1,-4,2,-3] => 4 [-1,-4,-2,3] => 4 [-1,-4,-2,-3] => 4 [1,4,3,2] => 2 [1,4,3,-2] => 2 [1,4,-3,2] => 2 [1,4,-3,-2] => 2 [1,-4,3,2] => 4 [1,-4,3,-2] => 4 [1,-4,-3,2] => 3 [1,-4,-3,-2] => 3 [-1,4,3,2] => 4 [-1,4,3,-2] => 4 [-1,4,-3,2] => 3 [-1,4,-3,-2] => 3 [-1,-4,3,2] => 5 [-1,-4,3,-2] => 5 [-1,-4,-3,2] => 3 [-1,-4,-3,-2] => 3 [2,1,3,4] => 2 [2,1,3,-4] => 4 [2,1,-3,4] => 4 [2,1,-3,-4] => 5 [2,-1,3,4] => 2 [2,-1,3,-4] => 4 [2,-1,-3,4] => 4 [2,-1,-3,-4] => 5 [-2,1,3,4] => 4 [-2,1,3,-4] => 5 [-2,1,-3,4] => 5 [-2,1,-3,-4] => 5 [-2,-1,3,4] => 4 [-2,-1,3,-4] => 5 [-2,-1,-3,4] => 5 [-2,-1,-3,-4] => 5 [2,1,4,3] => 2 [2,1,4,-3] => 3 [2,1,-4,3] => 3 [2,1,-4,-3] => 4 [2,-1,4,3] => 3 [2,-1,4,-3] => 3 [2,-1,-4,3] => 4 [2,-1,-4,-3] => 4 [-2,1,4,3] => 3 [-2,1,4,-3] => 4 [-2,1,-4,3] => 3 [-2,1,-4,-3] => 4 [-2,-1,4,3] => 4 [-2,-1,4,-3] => 4 [-2,-1,-4,3] => 4 [-2,-1,-4,-3] => 4 [2,3,1,4] => 1 [2,3,1,-4] => 3 [2,3,-1,4] => 1 [2,3,-1,-4] => 3 [2,-3,1,4] => 3 [2,-3,1,-4] => 4 [2,-3,-1,4] => 3 [2,-3,-1,-4] => 4 [-2,3,1,4] => 3 [-2,3,1,-4] => 4 [-2,3,-1,4] => 3 [-2,3,-1,-4] => 4 [-2,-3,1,4] => 4 [-2,-3,1,-4] => 4 [-2,-3,-1,4] => 4 [-2,-3,-1,-4] => 4 [2,3,4,1] => 0 [2,3,4,-1] => 0 [2,3,-4,1] => 2 [2,3,-4,-1] => 2 [2,-3,4,1] => 2 [2,-3,4,-1] => 2 [2,-3,-4,1] => 3 [2,-3,-4,-1] => 3 [-2,3,4,1] => 2 [-2,3,4,-1] => 2 [-2,3,-4,1] => 3 [-2,3,-4,-1] => 3 [-2,-3,4,1] => 3 [-2,-3,4,-1] => 3 [-2,-3,-4,1] => 3 [-2,-3,-4,-1] => 3 [2,4,1,3] => 1 [2,4,1,-3] => 2 [2,4,-1,3] => 2 [2,4,-1,-3] => 2 [2,-4,1,3] => 2 [2,-4,1,-3] => 3 [2,-4,-1,3] => 3 [2,-4,-1,-3] => 3 [-2,4,1,3] => 2 [-2,4,1,-3] => 3 [-2,4,-1,3] => 3 [-2,4,-1,-3] => 3 [-2,-4,1,3] => 2 [-2,-4,1,-3] => 3 [-2,-4,-1,3] => 3 [-2,-4,-1,-3] => 3 [2,4,3,1] => 1 [2,4,3,-1] => 1 [2,4,-3,1] => 1 [2,4,-3,-1] => 1 [2,-4,3,1] => 3 [2,-4,3,-1] => 3 [2,-4,-3,1] => 2 [2,-4,-3,-1] => 2 [-2,4,3,1] => 3 [-2,4,3,-1] => 3 [-2,4,-3,1] => 2 [-2,4,-3,-1] => 2 [-2,-4,3,1] => 4 [-2,-4,3,-1] => 4 [-2,-4,-3,1] => 2 [-2,-4,-3,-1] => 2 [3,1,2,4] => 2 [3,1,2,-4] => 3 [3,1,-2,4] => 3 [3,1,-2,-4] => 4 [3,-1,2,4] => 3 [3,-1,2,-4] => 4 [3,-1,-2,4] => 3 [3,-1,-2,-4] => 4 [-3,1,2,4] => 3 [-3,1,2,-4] => 3 [-3,1,-2,4] => 4 [-3,1,-2,-4] => 4 [-3,-1,2,4] => 4 [-3,-1,2,-4] => 4 [-3,-1,-2,4] => 4 [-3,-1,-2,-4] => 4 [3,1,4,2] => 1 [3,1,4,-2] => 2 [3,1,-4,2] => 2 [3,1,-4,-2] => 3 [3,-1,4,2] => 2 [3,-1,4,-2] => 2 [3,-1,-4,2] => 3 [3,-1,-4,-2] => 3 [-3,1,4,2] => 2 [-3,1,4,-2] => 3 [-3,1,-4,2] => 2 [-3,1,-4,-2] => 3 [-3,-1,4,2] => 3 [-3,-1,4,-2] => 3 [-3,-1,-4,2] => 3 [-3,-1,-4,-2] => 3 [3,2,1,4] => 2 [3,2,1,-4] => 4 [3,2,-1,4] => 2 [3,2,-1,-4] => 4 [3,-2,1,4] => 2 [3,-2,1,-4] => 3 [3,-2,-1,4] => 2 [3,-2,-1,-4] => 3 [-3,2,1,4] => 4 [-3,2,1,-4] => 5 [-3,2,-1,4] => 4 [-3,2,-1,-4] => 5 [-3,-2,1,4] => 3 [-3,-2,1,-4] => 3 [-3,-2,-1,4] => 3 [-3,-2,-1,-4] => 3 [3,2,4,1] => 1 [3,2,4,-1] => 1 [3,2,-4,1] => 3 [3,2,-4,-1] => 3 [3,-2,4,1] => 1 [3,-2,4,-1] => 1 [3,-2,-4,1] => 2 [3,-2,-4,-1] => 2 [-3,2,4,1] => 3 [-3,2,4,-1] => 3 [-3,2,-4,1] => 4 [-3,2,-4,-1] => 4 [-3,-2,4,1] => 2 [-3,-2,4,-1] => 2 [-3,-2,-4,1] => 2 [-3,-2,-4,-1] => 2 [3,4,1,2] => 0 [3,4,1,-2] => 1 [3,4,-1,2] => 1 [3,4,-1,-2] => 1 [3,-4,1,2] => 1 [3,-4,1,-2] => 2 [3,-4,-1,2] => 2 [3,-4,-1,-2] => 2 [-3,4,1,2] => 1 [-3,4,1,-2] => 2 [-3,4,-1,2] => 2 [-3,4,-1,-2] => 2 [-3,-4,1,2] => 1 [-3,-4,1,-2] => 2 [-3,-4,-1,2] => 2 [-3,-4,-1,-2] => 2 [3,4,2,1] => 1 [3,4,2,-1] => 2 [3,4,-2,1] => 0 [3,4,-2,-1] => 0 [3,-4,2,1] => 2 [3,-4,2,-1] => 3 [3,-4,-2,1] => 1 [3,-4,-2,-1] => 1 [-3,4,2,1] => 2 [-3,4,2,-1] => 3 [-3,4,-2,1] => 1 [-3,4,-2,-1] => 1 [-3,-4,2,1] => 2 [-3,-4,2,-1] => 3 [-3,-4,-2,1] => 1 [-3,-4,-2,-1] => 1 [4,1,2,3] => 0 [4,1,2,-3] => 2 [4,1,-2,3] => 2 [4,1,-2,-3] => 3 [4,-1,2,3] => 2 [4,-1,2,-3] => 3 [4,-1,-2,3] => 3 [4,-1,-2,-3] => 3 [-4,1,2,3] => 0 [-4,1,2,-3] => 2 [-4,1,-2,3] => 2 [-4,1,-2,-3] => 3 [-4,-1,2,3] => 2 [-4,-1,2,-3] => 3 [-4,-1,-2,3] => 3 [-4,-1,-2,-3] => 3 [4,1,3,2] => 2 [4,1,3,-2] => 3 [4,1,-3,2] => 1 [4,1,-3,-2] => 2 [4,-1,3,2] => 3 [4,-1,3,-2] => 3 [4,-1,-3,2] => 2 [4,-1,-3,-2] => 2 [-4,1,3,2] => 3 [-4,1,3,-2] => 4 [-4,1,-3,2] => 1 [-4,1,-3,-2] => 2 [-4,-1,3,2] => 4 [-4,-1,3,-2] => 4 [-4,-1,-3,2] => 2 [-4,-1,-3,-2] => 2 [4,2,1,3] => 2 [4,2,1,-3] => 3 [4,2,-1,3] => 3 [4,2,-1,-3] => 3 [4,-2,1,3] => 1 [4,-2,1,-3] => 2 [4,-2,-1,3] => 2 [4,-2,-1,-3] => 2 [-4,2,1,3] => 3 [-4,2,1,-3] => 4 [-4,2,-1,3] => 4 [-4,2,-1,-3] => 4 [-4,-2,1,3] => 1 [-4,-2,1,-3] => 2 [-4,-2,-1,3] => 2 [-4,-2,-1,-3] => 2 [4,2,3,1] => 2 [4,2,3,-1] => 2 [4,2,-3,1] => 2 [4,2,-3,-1] => 2 [4,-2,3,1] => 2 [4,-2,3,-1] => 2 [4,-2,-3,1] => 1 [4,-2,-3,-1] => 1 [-4,2,3,1] => 4 [-4,2,3,-1] => 4 [-4,2,-3,1] => 3 [-4,2,-3,-1] => 3 [-4,-2,3,1] => 3 [-4,-2,3,-1] => 3 [-4,-2,-3,1] => 1 [-4,-2,-3,-1] => 1 [4,3,1,2] => 1 [4,3,1,-2] => 2 [4,3,-1,2] => 2 [4,3,-1,-2] => 2 [4,-3,1,2] => 0 [4,-3,1,-2] => 1 [4,-3,-1,2] => 1 [4,-3,-1,-2] => 1 [-4,3,1,2] => 2 [-4,3,1,-2] => 3 [-4,3,-1,2] => 3 [-4,3,-1,-2] => 3 [-4,-3,1,2] => 0 [-4,-3,1,-2] => 1 [-4,-3,-1,2] => 1 [-4,-3,-1,-2] => 1 [4,3,2,1] => 2 [4,3,2,-1] => 3 [4,3,-2,1] => 1 [4,3,-2,-1] => 1 [4,-3,2,1] => 1 [4,-3,2,-1] => 2 [4,-3,-2,1] => 0 [4,-3,-2,-1] => 0 [-4,3,2,1] => 3 [-4,3,2,-1] => 4 [-4,3,-2,1] => 2 [-4,3,-2,-1] => 2 [-4,-3,2,1] => 1 [-4,-3,2,-1] => 2 [-4,-3,-2,1] => 0 [-4,-3,-2,-1] => 0 ----------------------------------------------------------------------------- Created: Jul 21, 2022 at 07:29 by Dennis Jahn ----------------------------------------------------------------------------- Last Updated: Dec 01, 2022 at 14:23 by Martin Rubey