***************************************************************************** * 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: St001823 ----------------------------------------------------------------------------- Collection: Signed permutations ----------------------------------------------------------------------------- Description: The Stasinski-Voll length of a signed permutation. The Stasinski-Voll length of a signed permutation $\sigma$ is $$L(\sigma) = \frac{1}{2} \#\{(i,j) ~\mid -n \leq i < j \leq n,~ i \not\equiv j \operatorname{mod} 2,~ \sigma(i) > \sigma(j)\},$$ where $n$ is the size of $\sigma$. ----------------------------------------------------------------------------- References: [1] Stasinski, A., Voll, C. A new statistic on the hyperoctahedral groups [[arXiv:1303.0990]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: [1] => 0 [-1] => 1 [1,2] => 0 [1,-2] => 2 [-1,2] => 1 [-1,-2] => 3 [2,1] => 1 [2,-1] => 1 [-2,1] => 2 [-2,-1] => 2 [1,2,3] => 0 [1,2,-3] => 3 [1,-2,3] => 2 [1,-2,-3] => 5 [-1,2,3] => 1 [-1,2,-3] => 4 [-1,-2,3] => 3 [-1,-2,-3] => 6 [1,3,2] => 1 [1,3,-2] => 2 [1,-3,2] => 3 [1,-3,-2] => 4 [-1,3,2] => 2 [-1,3,-2] => 3 [-1,-3,2] => 4 [-1,-3,-2] => 5 [2,1,3] => 1 [2,1,-3] => 4 [2,-1,3] => 1 [2,-1,-3] => 4 [-2,1,3] => 2 [-2,1,-3] => 5 [-2,-1,3] => 2 [-2,-1,-3] => 5 [2,3,1] => 1 [2,3,-1] => 2 [2,-3,1] => 3 [2,-3,-1] => 4 [-2,3,1] => 2 [-2,3,-1] => 3 [-2,-3,1] => 4 [-2,-3,-1] => 5 [3,1,2] => 1 [3,1,-2] => 4 [3,-1,2] => 1 [3,-1,-2] => 4 [-3,1,2] => 2 [-3,1,-2] => 5 [-3,-1,2] => 2 [-3,-1,-2] => 5 [3,2,1] => 2 [3,2,-1] => 3 [3,-2,1] => 2 [3,-2,-1] => 3 [-3,2,1] => 3 [-3,2,-1] => 4 [-3,-2,1] => 3 [-3,-2,-1] => 4 [1,2,3,4] => 0 [1,2,3,-4] => 4 [1,2,-3,4] => 3 [1,2,-3,-4] => 7 [1,-2,3,4] => 2 [1,-2,3,-4] => 6 [1,-2,-3,4] => 5 [1,-2,-3,-4] => 9 [-1,2,3,4] => 1 [-1,2,3,-4] => 5 [-1,2,-3,4] => 4 [-1,2,-3,-4] => 8 [-1,-2,3,4] => 3 [-1,-2,3,-4] => 7 [-1,-2,-3,4] => 6 [-1,-2,-3,-4] => 10 [1,2,4,3] => 1 [1,2,4,-3] => 3 [1,2,-4,3] => 4 [1,2,-4,-3] => 6 [1,-2,4,3] => 3 [1,-2,4,-3] => 5 [1,-2,-4,3] => 6 [1,-2,-4,-3] => 8 [-1,2,4,3] => 2 [-1,2,4,-3] => 4 [-1,2,-4,3] => 5 [-1,2,-4,-3] => 7 [-1,-2,4,3] => 4 [-1,-2,4,-3] => 6 [-1,-2,-4,3] => 7 [-1,-2,-4,-3] => 9 [1,3,2,4] => 1 [1,3,2,-4] => 5 [1,3,-2,4] => 2 [1,3,-2,-4] => 6 [1,-3,2,4] => 3 [1,-3,2,-4] => 7 [1,-3,-2,4] => 4 [1,-3,-2,-4] => 8 [-1,3,2,4] => 2 [-1,3,2,-4] => 6 [-1,3,-2,4] => 3 [-1,3,-2,-4] => 7 [-1,-3,2,4] => 4 [-1,-3,2,-4] => 8 [-1,-3,-2,4] => 5 [-1,-3,-2,-4] => 9 [1,3,4,2] => 1 [1,3,4,-2] => 3 [1,3,-4,2] => 4 [1,3,-4,-2] => 6 [1,-3,4,2] => 3 [1,-3,4,-2] => 5 [1,-3,-4,2] => 6 [1,-3,-4,-2] => 8 [-1,3,4,2] => 2 [-1,3,4,-2] => 4 [-1,3,-4,2] => 5 [-1,3,-4,-2] => 7 [-1,-3,4,2] => 4 [-1,-3,4,-2] => 6 [-1,-3,-4,2] => 7 [-1,-3,-4,-2] => 9 [1,4,2,3] => 1 [1,4,2,-3] => 5 [1,4,-2,3] => 2 [1,4,-2,-3] => 6 [1,-4,2,3] => 3 [1,-4,2,-3] => 7 [1,-4,-2,3] => 4 [1,-4,-2,-3] => 8 [-1,4,2,3] => 2 [-1,4,2,-3] => 6 [-1,4,-2,3] => 3 [-1,4,-2,-3] => 7 [-1,-4,2,3] => 4 [-1,-4,2,-3] => 8 [-1,-4,-2,3] => 5 [-1,-4,-2,-3] => 9 [1,4,3,2] => 2 [1,4,3,-2] => 4 [1,4,-3,2] => 3 [1,4,-3,-2] => 5 [1,-4,3,2] => 4 [1,-4,3,-2] => 6 [1,-4,-3,2] => 5 [1,-4,-3,-2] => 7 [-1,4,3,2] => 3 [-1,4,3,-2] => 5 [-1,4,-3,2] => 4 [-1,4,-3,-2] => 6 [-1,-4,3,2] => 5 [-1,-4,3,-2] => 7 [-1,-4,-3,2] => 6 [-1,-4,-3,-2] => 8 [2,1,3,4] => 1 [2,1,3,-4] => 5 [2,1,-3,4] => 4 [2,1,-3,-4] => 8 [2,-1,3,4] => 1 [2,-1,3,-4] => 5 [2,-1,-3,4] => 4 [2,-1,-3,-4] => 8 [-2,1,3,4] => 2 [-2,1,3,-4] => 6 [-2,1,-3,4] => 5 [-2,1,-3,-4] => 9 [-2,-1,3,4] => 2 [-2,-1,3,-4] => 6 [-2,-1,-3,4] => 5 [-2,-1,-3,-4] => 9 [2,1,4,3] => 2 [2,1,4,-3] => 4 [2,1,-4,3] => 5 [2,1,-4,-3] => 7 [2,-1,4,3] => 2 [2,-1,4,-3] => 4 [2,-1,-4,3] => 5 [2,-1,-4,-3] => 7 [-2,1,4,3] => 3 [-2,1,4,-3] => 5 [-2,1,-4,3] => 6 [-2,1,-4,-3] => 8 [-2,-1,4,3] => 3 [-2,-1,4,-3] => 5 [-2,-1,-4,3] => 6 [-2,-1,-4,-3] => 8 [2,3,1,4] => 1 [2,3,1,-4] => 5 [2,3,-1,4] => 2 [2,3,-1,-4] => 6 [2,-3,1,4] => 3 [2,-3,1,-4] => 7 [2,-3,-1,4] => 4 [2,-3,-1,-4] => 8 [-2,3,1,4] => 2 [-2,3,1,-4] => 6 [-2,3,-1,4] => 3 [-2,3,-1,-4] => 7 [-2,-3,1,4] => 4 [-2,-3,1,-4] => 8 [-2,-3,-1,4] => 5 [-2,-3,-1,-4] => 9 [2,3,4,1] => 2 [2,3,4,-1] => 2 [2,3,-4,1] => 5 [2,3,-4,-1] => 5 [2,-3,4,1] => 4 [2,-3,4,-1] => 4 [2,-3,-4,1] => 7 [2,-3,-4,-1] => 7 [-2,3,4,1] => 3 [-2,3,4,-1] => 3 [-2,3,-4,1] => 6 [-2,3,-4,-1] => 6 [-2,-3,4,1] => 5 [-2,-3,4,-1] => 5 [-2,-3,-4,1] => 8 [-2,-3,-4,-1] => 8 [2,4,1,3] => 1 [2,4,1,-3] => 5 [2,4,-1,3] => 2 [2,4,-1,-3] => 6 [2,-4,1,3] => 3 [2,-4,1,-3] => 7 [2,-4,-1,3] => 4 [2,-4,-1,-3] => 8 [-2,4,1,3] => 2 [-2,4,1,-3] => 6 [-2,4,-1,3] => 3 [-2,4,-1,-3] => 7 [-2,-4,1,3] => 4 [-2,-4,1,-3] => 8 [-2,-4,-1,3] => 5 [-2,-4,-1,-3] => 9 [2,4,3,1] => 3 [2,4,3,-1] => 3 [2,4,-3,1] => 4 [2,4,-3,-1] => 4 [2,-4,3,1] => 5 [2,-4,3,-1] => 5 [2,-4,-3,1] => 6 [2,-4,-3,-1] => 6 [-2,4,3,1] => 4 [-2,4,3,-1] => 4 [-2,4,-3,1] => 5 [-2,4,-3,-1] => 5 [-2,-4,3,1] => 6 [-2,-4,3,-1] => 6 [-2,-4,-3,1] => 7 [-2,-4,-3,-1] => 7 [3,1,2,4] => 1 [3,1,2,-4] => 5 [3,1,-2,4] => 4 [3,1,-2,-4] => 8 [3,-1,2,4] => 1 [3,-1,2,-4] => 5 [3,-1,-2,4] => 4 [3,-1,-2,-4] => 8 [-3,1,2,4] => 2 [-3,1,2,-4] => 6 [-3,1,-2,4] => 5 [-3,1,-2,-4] => 9 [-3,-1,2,4] => 2 [-3,-1,2,-4] => 6 [-3,-1,-2,4] => 5 [-3,-1,-2,-4] => 9 [3,1,4,2] => 3 [3,1,4,-2] => 3 [3,1,-4,2] => 6 [3,1,-4,-2] => 6 [3,-1,4,2] => 3 [3,-1,4,-2] => 3 [3,-1,-4,2] => 6 [3,-1,-4,-2] => 6 [-3,1,4,2] => 4 [-3,1,4,-2] => 4 [-3,1,-4,2] => 7 [-3,1,-4,-2] => 7 [-3,-1,4,2] => 4 [-3,-1,4,-2] => 4 [-3,-1,-4,2] => 7 [-3,-1,-4,-2] => 7 [3,2,1,4] => 2 [3,2,1,-4] => 6 [3,2,-1,4] => 3 [3,2,-1,-4] => 7 [3,-2,1,4] => 2 [3,-2,1,-4] => 6 [3,-2,-1,4] => 3 [3,-2,-1,-4] => 7 [-3,2,1,4] => 3 [-3,2,1,-4] => 7 [-3,2,-1,4] => 4 [-3,2,-1,-4] => 8 [-3,-2,1,4] => 3 [-3,-2,1,-4] => 7 [-3,-2,-1,4] => 4 [-3,-2,-1,-4] => 8 [3,2,4,1] => 3 [3,2,4,-1] => 3 [3,2,-4,1] => 6 [3,2,-4,-1] => 6 [3,-2,4,1] => 3 [3,-2,4,-1] => 3 [3,-2,-4,1] => 6 [3,-2,-4,-1] => 6 [-3,2,4,1] => 4 [-3,2,4,-1] => 4 [-3,2,-4,1] => 7 [-3,2,-4,-1] => 7 [-3,-2,4,1] => 4 [-3,-2,4,-1] => 4 [-3,-2,-4,1] => 7 [-3,-2,-4,-1] => 7 [3,4,1,2] => 2 [3,4,1,-2] => 4 [3,4,-1,2] => 3 [3,4,-1,-2] => 5 [3,-4,1,2] => 4 [3,-4,1,-2] => 6 [3,-4,-1,2] => 5 [3,-4,-1,-2] => 7 [-3,4,1,2] => 3 [-3,4,1,-2] => 5 [-3,4,-1,2] => 4 [-3,4,-1,-2] => 6 [-3,-4,1,2] => 5 [-3,-4,1,-2] => 7 [-3,-4,-1,2] => 6 [-3,-4,-1,-2] => 8 [3,4,2,1] => 3 [3,4,2,-1] => 3 [3,4,-2,1] => 4 [3,4,-2,-1] => 4 [3,-4,2,1] => 5 [3,-4,2,-1] => 5 [3,-4,-2,1] => 6 [3,-4,-2,-1] => 6 [-3,4,2,1] => 4 [-3,4,2,-1] => 4 [-3,4,-2,1] => 5 [-3,4,-2,-1] => 5 [-3,-4,2,1] => 6 [-3,-4,2,-1] => 6 [-3,-4,-2,1] => 7 [-3,-4,-2,-1] => 7 [4,1,2,3] => 2 [4,1,2,-3] => 4 [4,1,-2,3] => 5 [4,1,-2,-3] => 7 [4,-1,2,3] => 2 [4,-1,2,-3] => 4 [4,-1,-2,3] => 5 [4,-1,-2,-3] => 7 [-4,1,2,3] => 3 [-4,1,2,-3] => 5 [-4,1,-2,3] => 6 [-4,1,-2,-3] => 8 [-4,-1,2,3] => 3 [-4,-1,2,-3] => 5 [-4,-1,-2,3] => 6 [-4,-1,-2,-3] => 8 [4,1,3,2] => 3 [4,1,3,-2] => 3 [4,1,-3,2] => 6 [4,1,-3,-2] => 6 [4,-1,3,2] => 3 [4,-1,3,-2] => 3 [4,-1,-3,2] => 6 [4,-1,-3,-2] => 6 [-4,1,3,2] => 4 [-4,1,3,-2] => 4 [-4,1,-3,2] => 7 [-4,1,-3,-2] => 7 [-4,-1,3,2] => 4 [-4,-1,3,-2] => 4 [-4,-1,-3,2] => 7 [-4,-1,-3,-2] => 7 [4,2,1,3] => 3 [4,2,1,-3] => 5 [4,2,-1,3] => 4 [4,2,-1,-3] => 6 [4,-2,1,3] => 3 [4,-2,1,-3] => 5 [4,-2,-1,3] => 4 [4,-2,-1,-3] => 6 [-4,2,1,3] => 4 [-4,2,1,-3] => 6 [-4,2,-1,3] => 5 [-4,2,-1,-3] => 7 [-4,-2,1,3] => 4 [-4,-2,1,-3] => 6 [-4,-2,-1,3] => 5 [-4,-2,-1,-3] => 7 [4,2,3,1] => 3 [4,2,3,-1] => 3 [4,2,-3,1] => 6 [4,2,-3,-1] => 6 [4,-2,3,1] => 3 [4,-2,3,-1] => 3 [4,-2,-3,1] => 6 [4,-2,-3,-1] => 6 [-4,2,3,1] => 4 [-4,2,3,-1] => 4 [-4,2,-3,1] => 7 [-4,2,-3,-1] => 7 [-4,-2,3,1] => 4 [-4,-2,3,-1] => 4 [-4,-2,-3,1] => 7 [-4,-2,-3,-1] => 7 [4,3,1,2] => 3 [4,3,1,-2] => 5 [4,3,-1,2] => 4 [4,3,-1,-2] => 6 [4,-3,1,2] => 3 [4,-3,1,-2] => 5 [4,-3,-1,2] => 4 [4,-3,-1,-2] => 6 [-4,3,1,2] => 4 [-4,3,1,-2] => 6 [-4,3,-1,2] => 5 [-4,3,-1,-2] => 7 [-4,-3,1,2] => 4 [-4,-3,1,-2] => 6 [-4,-3,-1,2] => 5 [-4,-3,-1,-2] => 7 [4,3,2,1] => 4 [4,3,2,-1] => 4 [4,3,-2,1] => 5 [4,3,-2,-1] => 5 [4,-3,2,1] => 4 [4,-3,2,-1] => 4 [4,-3,-2,1] => 5 [4,-3,-2,-1] => 5 [-4,3,2,1] => 5 [-4,3,2,-1] => 5 [-4,3,-2,1] => 6 [-4,3,-2,-1] => 6 [-4,-3,2,1] => 5 [-4,-3,2,-1] => 5 [-4,-3,-2,1] => 6 [-4,-3,-2,-1] => 6 ----------------------------------------------------------------------------- Created: Jul 22, 2022 at 11:45 by Dennis Jahn ----------------------------------------------------------------------------- Last Updated: Jul 22, 2022 at 11:45 by Dennis Jahn