***************************************************************************** * 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: St001433 ----------------------------------------------------------------------------- Collection: Signed permutations ----------------------------------------------------------------------------- Description: The flag major index of a signed permutation. The flag major index of a signed permutation $\sigma$ is: $$\operatorname{fmaj}(\sigma)=\operatorname{neg}(\sigma)+2\cdot \sum_{i\in \operatorname{Des}_B(\sigma)}{i} ,$$ where $\operatorname{Des}_B(\sigma)$ is the $B$-descent set of $\sigma$; see [1, Eq.(10)]. This statistic is equidistributed with the $B$-inversions ([[St001428]]) and with the negative major index on the groups of signed permutations (see [1, Corollary 4.6]). ----------------------------------------------------------------------------- References: [1] Adinh, R. M., Brenti, F., Roichman, Y. Descent numbers and major indices for the hyperoctahedral group [[MathSciNet:2003m:05211]] [[arXiv:math/0012111]] ----------------------------------------------------------------------------- Code: def statistic(pi): pi = [0] + list(pi) return (sum(1 for i in range(len(pi)) if pi[i] < 0) + sum(2*i for i in range(len(pi)-1) if pi[i] > pi[i+1])) ----------------------------------------------------------------------------- Statistic values: [1] => 0 [-1] => 1 [1,2] => 0 [1,-2] => 3 [-1,2] => 1 [-1,-2] => 4 [2,1] => 2 [2,-1] => 3 [-2,1] => 1 [-2,-1] => 2 [1,2,3] => 0 [1,2,-3] => 5 [1,-2,3] => 3 [1,-2,-3] => 8 [-1,2,3] => 1 [-1,2,-3] => 6 [-1,-2,3] => 4 [-1,-2,-3] => 9 [1,3,2] => 4 [1,3,-2] => 5 [1,-3,2] => 3 [1,-3,-2] => 4 [-1,3,2] => 5 [-1,3,-2] => 6 [-1,-3,2] => 4 [-1,-3,-2] => 5 [2,1,3] => 2 [2,1,-3] => 7 [2,-1,3] => 3 [2,-1,-3] => 8 [-2,1,3] => 1 [-2,1,-3] => 6 [-2,-1,3] => 2 [-2,-1,-3] => 7 [2,3,1] => 4 [2,3,-1] => 5 [2,-3,1] => 3 [2,-3,-1] => 4 [-2,3,1] => 5 [-2,3,-1] => 6 [-2,-3,1] => 4 [-2,-3,-1] => 5 [3,1,2] => 2 [3,1,-2] => 7 [3,-1,2] => 3 [3,-1,-2] => 8 [-3,1,2] => 1 [-3,1,-2] => 6 [-3,-1,2] => 2 [-3,-1,-2] => 7 [3,2,1] => 6 [3,2,-1] => 7 [3,-2,1] => 3 [3,-2,-1] => 4 [-3,2,1] => 5 [-3,2,-1] => 6 [-3,-2,1] => 2 [-3,-2,-1] => 3 [1,2,3,4] => 0 [1,2,3,-4] => 7 [1,2,-3,4] => 5 [1,2,-3,-4] => 12 [1,-2,3,4] => 3 [1,-2,3,-4] => 10 [1,-2,-3,4] => 8 [1,-2,-3,-4] => 15 [-1,2,3,4] => 1 [-1,2,3,-4] => 8 [-1,2,-3,4] => 6 [-1,2,-3,-4] => 13 [-1,-2,3,4] => 4 [-1,-2,3,-4] => 11 [-1,-2,-3,4] => 9 [-1,-2,-3,-4] => 16 [1,2,4,3] => 6 [1,2,4,-3] => 7 [1,2,-4,3] => 5 [1,2,-4,-3] => 6 [1,-2,4,3] => 9 [1,-2,4,-3] => 10 [1,-2,-4,3] => 8 [1,-2,-4,-3] => 9 [-1,2,4,3] => 7 [-1,2,4,-3] => 8 [-1,2,-4,3] => 6 [-1,2,-4,-3] => 7 [-1,-2,4,3] => 10 [-1,-2,4,-3] => 11 [-1,-2,-4,3] => 9 [-1,-2,-4,-3] => 10 [1,3,2,4] => 4 [1,3,2,-4] => 11 [1,3,-2,4] => 5 [1,3,-2,-4] => 12 [1,-3,2,4] => 3 [1,-3,2,-4] => 10 [1,-3,-2,4] => 4 [1,-3,-2,-4] => 11 [-1,3,2,4] => 5 [-1,3,2,-4] => 12 [-1,3,-2,4] => 6 [-1,3,-2,-4] => 13 [-1,-3,2,4] => 4 [-1,-3,2,-4] => 11 [-1,-3,-2,4] => 5 [-1,-3,-2,-4] => 12 [1,3,4,2] => 6 [1,3,4,-2] => 7 [1,3,-4,2] => 5 [1,3,-4,-2] => 6 [1,-3,4,2] => 9 [1,-3,4,-2] => 10 [1,-3,-4,2] => 8 [1,-3,-4,-2] => 9 [-1,3,4,2] => 7 [-1,3,4,-2] => 8 [-1,3,-4,2] => 6 [-1,3,-4,-2] => 7 [-1,-3,4,2] => 10 [-1,-3,4,-2] => 11 [-1,-3,-4,2] => 9 [-1,-3,-4,-2] => 10 [1,4,2,3] => 4 [1,4,2,-3] => 11 [1,4,-2,3] => 5 [1,4,-2,-3] => 12 [1,-4,2,3] => 3 [1,-4,2,-3] => 10 [1,-4,-2,3] => 4 [1,-4,-2,-3] => 11 [-1,4,2,3] => 5 [-1,4,2,-3] => 12 [-1,4,-2,3] => 6 [-1,4,-2,-3] => 13 [-1,-4,2,3] => 4 [-1,-4,2,-3] => 11 [-1,-4,-2,3] => 5 [-1,-4,-2,-3] => 12 [1,4,3,2] => 10 [1,4,3,-2] => 11 [1,4,-3,2] => 5 [1,4,-3,-2] => 6 [1,-4,3,2] => 9 [1,-4,3,-2] => 10 [1,-4,-3,2] => 4 [1,-4,-3,-2] => 5 [-1,4,3,2] => 11 [-1,4,3,-2] => 12 [-1,4,-3,2] => 6 [-1,4,-3,-2] => 7 [-1,-4,3,2] => 10 [-1,-4,3,-2] => 11 [-1,-4,-3,2] => 5 [-1,-4,-3,-2] => 6 [2,1,3,4] => 2 [2,1,3,-4] => 9 [2,1,-3,4] => 7 [2,1,-3,-4] => 14 [2,-1,3,4] => 3 [2,-1,3,-4] => 10 [2,-1,-3,4] => 8 [2,-1,-3,-4] => 15 [-2,1,3,4] => 1 [-2,1,3,-4] => 8 [-2,1,-3,4] => 6 [-2,1,-3,-4] => 13 [-2,-1,3,4] => 2 [-2,-1,3,-4] => 9 [-2,-1,-3,4] => 7 [-2,-1,-3,-4] => 14 [2,1,4,3] => 8 [2,1,4,-3] => 9 [2,1,-4,3] => 7 [2,1,-4,-3] => 8 [2,-1,4,3] => 9 [2,-1,4,-3] => 10 [2,-1,-4,3] => 8 [2,-1,-4,-3] => 9 [-2,1,4,3] => 7 [-2,1,4,-3] => 8 [-2,1,-4,3] => 6 [-2,1,-4,-3] => 7 [-2,-1,4,3] => 8 [-2,-1,4,-3] => 9 [-2,-1,-4,3] => 7 [-2,-1,-4,-3] => 8 [2,3,1,4] => 4 [2,3,1,-4] => 11 [2,3,-1,4] => 5 [2,3,-1,-4] => 12 [2,-3,1,4] => 3 [2,-3,1,-4] => 10 [2,-3,-1,4] => 4 [2,-3,-1,-4] => 11 [-2,3,1,4] => 5 [-2,3,1,-4] => 12 [-2,3,-1,4] => 6 [-2,3,-1,-4] => 13 [-2,-3,1,4] => 4 [-2,-3,1,-4] => 11 [-2,-3,-1,4] => 5 [-2,-3,-1,-4] => 12 [2,3,4,1] => 6 [2,3,4,-1] => 7 [2,3,-4,1] => 5 [2,3,-4,-1] => 6 [2,-3,4,1] => 9 [2,-3,4,-1] => 10 [2,-3,-4,1] => 8 [2,-3,-4,-1] => 9 [-2,3,4,1] => 7 [-2,3,4,-1] => 8 [-2,3,-4,1] => 6 [-2,3,-4,-1] => 7 [-2,-3,4,1] => 10 [-2,-3,4,-1] => 11 [-2,-3,-4,1] => 9 [-2,-3,-4,-1] => 10 [2,4,1,3] => 4 [2,4,1,-3] => 11 [2,4,-1,3] => 5 [2,4,-1,-3] => 12 [2,-4,1,3] => 3 [2,-4,1,-3] => 10 [2,-4,-1,3] => 4 [2,-4,-1,-3] => 11 [-2,4,1,3] => 5 [-2,4,1,-3] => 12 [-2,4,-1,3] => 6 [-2,4,-1,-3] => 13 [-2,-4,1,3] => 4 [-2,-4,1,-3] => 11 [-2,-4,-1,3] => 5 [-2,-4,-1,-3] => 12 [2,4,3,1] => 10 [2,4,3,-1] => 11 [2,4,-3,1] => 5 [2,4,-3,-1] => 6 [2,-4,3,1] => 9 [2,-4,3,-1] => 10 [2,-4,-3,1] => 4 [2,-4,-3,-1] => 5 [-2,4,3,1] => 11 [-2,4,3,-1] => 12 [-2,4,-3,1] => 6 [-2,4,-3,-1] => 7 [-2,-4,3,1] => 10 [-2,-4,3,-1] => 11 [-2,-4,-3,1] => 5 [-2,-4,-3,-1] => 6 [3,1,2,4] => 2 [3,1,2,-4] => 9 [3,1,-2,4] => 7 [3,1,-2,-4] => 14 [3,-1,2,4] => 3 [3,-1,2,-4] => 10 [3,-1,-2,4] => 8 [3,-1,-2,-4] => 15 [-3,1,2,4] => 1 [-3,1,2,-4] => 8 [-3,1,-2,4] => 6 [-3,1,-2,-4] => 13 [-3,-1,2,4] => 2 [-3,-1,2,-4] => 9 [-3,-1,-2,4] => 7 [-3,-1,-2,-4] => 14 [3,1,4,2] => 8 [3,1,4,-2] => 9 [3,1,-4,2] => 7 [3,1,-4,-2] => 8 [3,-1,4,2] => 9 [3,-1,4,-2] => 10 [3,-1,-4,2] => 8 [3,-1,-4,-2] => 9 [-3,1,4,2] => 7 [-3,1,4,-2] => 8 [-3,1,-4,2] => 6 [-3,1,-4,-2] => 7 [-3,-1,4,2] => 8 [-3,-1,4,-2] => 9 [-3,-1,-4,2] => 7 [-3,-1,-4,-2] => 8 [3,2,1,4] => 6 [3,2,1,-4] => 13 [3,2,-1,4] => 7 [3,2,-1,-4] => 14 [3,-2,1,4] => 3 [3,-2,1,-4] => 10 [3,-2,-1,4] => 4 [3,-2,-1,-4] => 11 [-3,2,1,4] => 5 [-3,2,1,-4] => 12 [-3,2,-1,4] => 6 [-3,2,-1,-4] => 13 [-3,-2,1,4] => 2 [-3,-2,1,-4] => 9 [-3,-2,-1,4] => 3 [-3,-2,-1,-4] => 10 [3,2,4,1] => 8 [3,2,4,-1] => 9 [3,2,-4,1] => 7 [3,2,-4,-1] => 8 [3,-2,4,1] => 9 [3,-2,4,-1] => 10 [3,-2,-4,1] => 8 [3,-2,-4,-1] => 9 [-3,2,4,1] => 7 [-3,2,4,-1] => 8 [-3,2,-4,1] => 6 [-3,2,-4,-1] => 7 [-3,-2,4,1] => 8 [-3,-2,4,-1] => 9 [-3,-2,-4,1] => 7 [-3,-2,-4,-1] => 8 [3,4,1,2] => 4 [3,4,1,-2] => 11 [3,4,-1,2] => 5 [3,4,-1,-2] => 12 [3,-4,1,2] => 3 [3,-4,1,-2] => 10 [3,-4,-1,2] => 4 [3,-4,-1,-2] => 11 [-3,4,1,2] => 5 [-3,4,1,-2] => 12 [-3,4,-1,2] => 6 [-3,4,-1,-2] => 13 [-3,-4,1,2] => 4 [-3,-4,1,-2] => 11 [-3,-4,-1,2] => 5 [-3,-4,-1,-2] => 12 [3,4,2,1] => 10 [3,4,2,-1] => 11 [3,4,-2,1] => 5 [3,4,-2,-1] => 6 [3,-4,2,1] => 9 [3,-4,2,-1] => 10 [3,-4,-2,1] => 4 [3,-4,-2,-1] => 5 [-3,4,2,1] => 11 [-3,4,2,-1] => 12 [-3,4,-2,1] => 6 [-3,4,-2,-1] => 7 [-3,-4,2,1] => 10 [-3,-4,2,-1] => 11 [-3,-4,-2,1] => 5 [-3,-4,-2,-1] => 6 [4,1,2,3] => 2 [4,1,2,-3] => 9 [4,1,-2,3] => 7 [4,1,-2,-3] => 14 [4,-1,2,3] => 3 [4,-1,2,-3] => 10 [4,-1,-2,3] => 8 [4,-1,-2,-3] => 15 [-4,1,2,3] => 1 [-4,1,2,-3] => 8 [-4,1,-2,3] => 6 [-4,1,-2,-3] => 13 [-4,-1,2,3] => 2 [-4,-1,2,-3] => 9 [-4,-1,-2,3] => 7 [-4,-1,-2,-3] => 14 [4,1,3,2] => 8 [4,1,3,-2] => 9 [4,1,-3,2] => 7 [4,1,-3,-2] => 8 [4,-1,3,2] => 9 [4,-1,3,-2] => 10 [4,-1,-3,2] => 8 [4,-1,-3,-2] => 9 [-4,1,3,2] => 7 [-4,1,3,-2] => 8 [-4,1,-3,2] => 6 [-4,1,-3,-2] => 7 [-4,-1,3,2] => 8 [-4,-1,3,-2] => 9 [-4,-1,-3,2] => 7 [-4,-1,-3,-2] => 8 [4,2,1,3] => 6 [4,2,1,-3] => 13 [4,2,-1,3] => 7 [4,2,-1,-3] => 14 [4,-2,1,3] => 3 [4,-2,1,-3] => 10 [4,-2,-1,3] => 4 [4,-2,-1,-3] => 11 [-4,2,1,3] => 5 [-4,2,1,-3] => 12 [-4,2,-1,3] => 6 [-4,2,-1,-3] => 13 [-4,-2,1,3] => 2 [-4,-2,1,-3] => 9 [-4,-2,-1,3] => 3 [-4,-2,-1,-3] => 10 [4,2,3,1] => 8 [4,2,3,-1] => 9 [4,2,-3,1] => 7 [4,2,-3,-1] => 8 [4,-2,3,1] => 9 [4,-2,3,-1] => 10 [4,-2,-3,1] => 8 [4,-2,-3,-1] => 9 [-4,2,3,1] => 7 [-4,2,3,-1] => 8 [-4,2,-3,1] => 6 [-4,2,-3,-1] => 7 [-4,-2,3,1] => 8 [-4,-2,3,-1] => 9 [-4,-2,-3,1] => 7 [-4,-2,-3,-1] => 8 [4,3,1,2] => 6 [4,3,1,-2] => 13 [4,3,-1,2] => 7 [4,3,-1,-2] => 14 [4,-3,1,2] => 3 [4,-3,1,-2] => 10 [4,-3,-1,2] => 4 [4,-3,-1,-2] => 11 [-4,3,1,2] => 5 [-4,3,1,-2] => 12 [-4,3,-1,2] => 6 [-4,3,-1,-2] => 13 [-4,-3,1,2] => 2 [-4,-3,1,-2] => 9 [-4,-3,-1,2] => 3 [-4,-3,-1,-2] => 10 [4,3,2,1] => 12 [4,3,2,-1] => 13 [4,3,-2,1] => 7 [4,3,-2,-1] => 8 [4,-3,2,1] => 9 [4,-3,2,-1] => 10 [4,-3,-2,1] => 4 [4,-3,-2,-1] => 5 [-4,3,2,1] => 11 [-4,3,2,-1] => 12 [-4,3,-2,1] => 6 [-4,3,-2,-1] => 7 [-4,-3,2,1] => 8 [-4,-3,2,-1] => 9 [-4,-3,-2,1] => 3 [-4,-3,-2,-1] => 4 ----------------------------------------------------------------------------- Created: Jun 24, 2019 at 15:21 by Angela Carnevale ----------------------------------------------------------------------------- Last Updated: Jan 08, 2023 at 15:10 by Martin Rubey