***************************************************************************** * 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: St001819 ----------------------------------------------------------------------------- Collection: Signed permutations ----------------------------------------------------------------------------- Description: The flag Denert index of a signed permutation. The flag Denert index of a signed permutation $\sigma$ is: $$fden(\sigma) = \operatorname{neg}(\sigma) + 2 \cdot den_B(\sigma),$$ where $den_B(\sigma) = den(perm(\sigma))$ is the Denert index of the associated permutation to $\sigma$. ----------------------------------------------------------------------------- References: [1] Fire, M. Statistics on Wreath Products [[arXiv:math/0409421]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: [1] => 0 [-1] => 1 [1,2] => 0 [1,-2] => 1 [-1,2] => 1 [-1,-2] => 2 [2,1] => 2 [2,-1] => 3 [-2,1] => 3 [-2,-1] => 4 [1,2,3] => 0 [1,2,-3] => 1 [1,-2,3] => 1 [1,-2,-3] => 2 [-1,2,3] => 1 [-1,2,-3] => 2 [-1,-2,3] => 2 [-1,-2,-3] => 3 [1,3,2] => 4 [1,3,-2] => 5 [1,-3,2] => 5 [1,-3,-2] => 6 [-1,3,2] => 5 [-1,3,-2] => 6 [-1,-3,2] => 6 [-1,-3,-2] => 7 [2,1,3] => 2 [2,1,-3] => 3 [2,-1,3] => 3 [2,-1,-3] => 4 [-2,1,3] => 3 [-2,1,-3] => 4 [-2,-1,3] => 4 [-2,-1,-3] => 5 [2,3,1] => 6 [2,3,-1] => 7 [2,-3,1] => 7 [2,-3,-1] => 8 [-2,3,1] => 7 [-2,3,-1] => 8 [-2,-3,1] => 8 [-2,-3,-1] => 9 [3,1,2] => 2 [3,1,-2] => 3 [3,-1,2] => 3 [3,-1,-2] => 4 [-3,1,2] => 3 [-3,1,-2] => 4 [-3,-1,2] => 4 [-3,-1,-2] => 5 [3,2,1] => 4 [3,2,-1] => 5 [3,-2,1] => 5 [3,-2,-1] => 6 [-3,2,1] => 5 [-3,2,-1] => 6 [-3,-2,1] => 6 [-3,-2,-1] => 7 [1,2,3,4] => 0 [1,2,3,-4] => 1 [1,2,-3,4] => 1 [1,2,-3,-4] => 2 [1,-2,3,4] => 1 [1,-2,3,-4] => 2 [1,-2,-3,4] => 2 [1,-2,-3,-4] => 3 [-1,2,3,4] => 1 [-1,2,3,-4] => 2 [-1,2,-3,4] => 2 [-1,2,-3,-4] => 3 [-1,-2,3,4] => 2 [-1,-2,3,-4] => 3 [-1,-2,-3,4] => 3 [-1,-2,-3,-4] => 4 [1,2,4,3] => 6 [1,2,4,-3] => 7 [1,2,-4,3] => 7 [1,2,-4,-3] => 8 [1,-2,4,3] => 7 [1,-2,4,-3] => 8 [1,-2,-4,3] => 8 [1,-2,-4,-3] => 9 [-1,2,4,3] => 7 [-1,2,4,-3] => 8 [-1,2,-4,3] => 8 [-1,2,-4,-3] => 9 [-1,-2,4,3] => 8 [-1,-2,4,-3] => 9 [-1,-2,-4,3] => 9 [-1,-2,-4,-3] => 10 [1,3,2,4] => 4 [1,3,2,-4] => 5 [1,3,-2,4] => 5 [1,3,-2,-4] => 6 [1,-3,2,4] => 5 [1,-3,2,-4] => 6 [1,-3,-2,4] => 6 [1,-3,-2,-4] => 7 [-1,3,2,4] => 5 [-1,3,2,-4] => 6 [-1,3,-2,4] => 6 [-1,3,-2,-4] => 7 [-1,-3,2,4] => 6 [-1,-3,2,-4] => 7 [-1,-3,-2,4] => 7 [-1,-3,-2,-4] => 8 [1,3,4,2] => 10 [1,3,4,-2] => 11 [1,3,-4,2] => 11 [1,3,-4,-2] => 12 [1,-3,4,2] => 11 [1,-3,4,-2] => 12 [1,-3,-4,2] => 12 [1,-3,-4,-2] => 13 [-1,3,4,2] => 11 [-1,3,4,-2] => 12 [-1,3,-4,2] => 12 [-1,3,-4,-2] => 13 [-1,-3,4,2] => 12 [-1,-3,4,-2] => 13 [-1,-3,-4,2] => 13 [-1,-3,-4,-2] => 14 [1,4,2,3] => 4 [1,4,2,-3] => 5 [1,4,-2,3] => 5 [1,4,-2,-3] => 6 [1,-4,2,3] => 5 [1,-4,2,-3] => 6 [1,-4,-2,3] => 6 [1,-4,-2,-3] => 7 [-1,4,2,3] => 5 [-1,4,2,-3] => 6 [-1,4,-2,3] => 6 [-1,4,-2,-3] => 7 [-1,-4,2,3] => 6 [-1,-4,2,-3] => 7 [-1,-4,-2,3] => 7 [-1,-4,-2,-3] => 8 [1,4,3,2] => 6 [1,4,3,-2] => 7 [1,4,-3,2] => 7 [1,4,-3,-2] => 8 [1,-4,3,2] => 7 [1,-4,3,-2] => 8 [1,-4,-3,2] => 8 [1,-4,-3,-2] => 9 [-1,4,3,2] => 7 [-1,4,3,-2] => 8 [-1,4,-3,2] => 8 [-1,4,-3,-2] => 9 [-1,-4,3,2] => 8 [-1,-4,3,-2] => 9 [-1,-4,-3,2] => 9 [-1,-4,-3,-2] => 10 [2,1,3,4] => 2 [2,1,3,-4] => 3 [2,1,-3,4] => 3 [2,1,-3,-4] => 4 [2,-1,3,4] => 3 [2,-1,3,-4] => 4 [2,-1,-3,4] => 4 [2,-1,-3,-4] => 5 [-2,1,3,4] => 3 [-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] => 6 [2,1,4,3] => 8 [2,1,4,-3] => 9 [2,1,-4,3] => 9 [2,1,-4,-3] => 10 [2,-1,4,3] => 9 [2,-1,4,-3] => 10 [2,-1,-4,3] => 10 [2,-1,-4,-3] => 11 [-2,1,4,3] => 9 [-2,1,4,-3] => 10 [-2,1,-4,3] => 10 [-2,1,-4,-3] => 11 [-2,-1,4,3] => 10 [-2,-1,4,-3] => 11 [-2,-1,-4,3] => 11 [-2,-1,-4,-3] => 12 [2,3,1,4] => 6 [2,3,1,-4] => 7 [2,3,-1,4] => 7 [2,3,-1,-4] => 8 [2,-3,1,4] => 7 [2,-3,1,-4] => 8 [2,-3,-1,4] => 8 [2,-3,-1,-4] => 9 [-2,3,1,4] => 7 [-2,3,1,-4] => 8 [-2,3,-1,4] => 8 [-2,3,-1,-4] => 9 [-2,-3,1,4] => 8 [-2,-3,1,-4] => 9 [-2,-3,-1,4] => 9 [-2,-3,-1,-4] => 10 [2,3,4,1] => 12 [2,3,4,-1] => 13 [2,3,-4,1] => 13 [2,3,-4,-1] => 14 [2,-3,4,1] => 13 [2,-3,4,-1] => 14 [2,-3,-4,1] => 14 [2,-3,-4,-1] => 15 [-2,3,4,1] => 13 [-2,3,4,-1] => 14 [-2,3,-4,1] => 14 [-2,3,-4,-1] => 15 [-2,-3,4,1] => 14 [-2,-3,4,-1] => 15 [-2,-3,-4,1] => 15 [-2,-3,-4,-1] => 16 [2,4,1,3] => 6 [2,4,1,-3] => 7 [2,4,-1,3] => 7 [2,4,-1,-3] => 8 [2,-4,1,3] => 7 [2,-4,1,-3] => 8 [2,-4,-1,3] => 8 [2,-4,-1,-3] => 9 [-2,4,1,3] => 7 [-2,4,1,-3] => 8 [-2,4,-1,3] => 8 [-2,4,-1,-3] => 9 [-2,-4,1,3] => 8 [-2,-4,1,-3] => 9 [-2,-4,-1,3] => 9 [-2,-4,-1,-3] => 10 [2,4,3,1] => 8 [2,4,3,-1] => 9 [2,4,-3,1] => 9 [2,4,-3,-1] => 10 [2,-4,3,1] => 9 [2,-4,3,-1] => 10 [2,-4,-3,1] => 10 [2,-4,-3,-1] => 11 [-2,4,3,1] => 9 [-2,4,3,-1] => 10 [-2,4,-3,1] => 10 [-2,4,-3,-1] => 11 [-2,-4,3,1] => 10 [-2,-4,3,-1] => 11 [-2,-4,-3,1] => 11 [-2,-4,-3,-1] => 12 [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] => 4 [3,-1,-2,-4] => 5 [-3,1,2,4] => 3 [-3,1,2,-4] => 4 [-3,1,-2,4] => 4 [-3,1,-2,-4] => 5 [-3,-1,2,4] => 4 [-3,-1,2,-4] => 5 [-3,-1,-2,4] => 5 [-3,-1,-2,-4] => 6 [3,1,4,2] => 8 [3,1,4,-2] => 9 [3,1,-4,2] => 9 [3,1,-4,-2] => 10 [3,-1,4,2] => 9 [3,-1,4,-2] => 10 [3,-1,-4,2] => 10 [3,-1,-4,-2] => 11 [-3,1,4,2] => 9 [-3,1,4,-2] => 10 [-3,1,-4,2] => 10 [-3,1,-4,-2] => 11 [-3,-1,4,2] => 10 [-3,-1,4,-2] => 11 [-3,-1,-4,2] => 11 [-3,-1,-4,-2] => 12 [3,2,1,4] => 4 [3,2,1,-4] => 5 [3,2,-1,4] => 5 [3,2,-1,-4] => 6 [3,-2,1,4] => 5 [3,-2,1,-4] => 6 [3,-2,-1,4] => 6 [3,-2,-1,-4] => 7 [-3,2,1,4] => 5 [-3,2,1,-4] => 6 [-3,2,-1,4] => 6 [-3,2,-1,-4] => 7 [-3,-2,1,4] => 6 [-3,-2,1,-4] => 7 [-3,-2,-1,4] => 7 [-3,-2,-1,-4] => 8 [3,2,4,1] => 10 [3,2,4,-1] => 11 [3,2,-4,1] => 11 [3,2,-4,-1] => 12 [3,-2,4,1] => 11 [3,-2,4,-1] => 12 [3,-2,-4,1] => 12 [3,-2,-4,-1] => 13 [-3,2,4,1] => 11 [-3,2,4,-1] => 12 [-3,2,-4,1] => 12 [-3,2,-4,-1] => 13 [-3,-2,4,1] => 12 [-3,-2,4,-1] => 13 [-3,-2,-4,1] => 13 [-3,-2,-4,-1] => 14 [3,4,1,2] => 6 [3,4,1,-2] => 7 [3,4,-1,2] => 7 [3,4,-1,-2] => 8 [3,-4,1,2] => 7 [3,-4,1,-2] => 8 [3,-4,-1,2] => 8 [3,-4,-1,-2] => 9 [-3,4,1,2] => 7 [-3,4,1,-2] => 8 [-3,4,-1,2] => 8 [-3,4,-1,-2] => 9 [-3,-4,1,2] => 8 [-3,-4,1,-2] => 9 [-3,-4,-1,2] => 9 [-3,-4,-1,-2] => 10 [3,4,2,1] => 8 [3,4,2,-1] => 9 [3,4,-2,1] => 9 [3,4,-2,-1] => 10 [3,-4,2,1] => 9 [3,-4,2,-1] => 10 [3,-4,-2,1] => 10 [3,-4,-2,-1] => 11 [-3,4,2,1] => 9 [-3,4,2,-1] => 10 [-3,4,-2,1] => 10 [-3,4,-2,-1] => 11 [-3,-4,2,1] => 10 [-3,-4,2,-1] => 11 [-3,-4,-2,1] => 11 [-3,-4,-2,-1] => 12 [4,1,2,3] => 2 [4,1,2,-3] => 3 [4,1,-2,3] => 3 [4,1,-2,-3] => 4 [4,-1,2,3] => 3 [4,-1,2,-3] => 4 [4,-1,-2,3] => 4 [4,-1,-2,-3] => 5 [-4,1,2,3] => 3 [-4,1,2,-3] => 4 [-4,1,-2,3] => 4 [-4,1,-2,-3] => 5 [-4,-1,2,3] => 4 [-4,-1,2,-3] => 5 [-4,-1,-2,3] => 5 [-4,-1,-2,-3] => 6 [4,1,3,2] => 4 [4,1,3,-2] => 5 [4,1,-3,2] => 5 [4,1,-3,-2] => 6 [4,-1,3,2] => 5 [4,-1,3,-2] => 6 [4,-1,-3,2] => 6 [4,-1,-3,-2] => 7 [-4,1,3,2] => 5 [-4,1,3,-2] => 6 [-4,1,-3,2] => 6 [-4,1,-3,-2] => 7 [-4,-1,3,2] => 6 [-4,-1,3,-2] => 7 [-4,-1,-3,2] => 7 [-4,-1,-3,-2] => 8 [4,2,1,3] => 4 [4,2,1,-3] => 5 [4,2,-1,3] => 5 [4,2,-1,-3] => 6 [4,-2,1,3] => 5 [4,-2,1,-3] => 6 [4,-2,-1,3] => 6 [4,-2,-1,-3] => 7 [-4,2,1,3] => 5 [-4,2,1,-3] => 6 [-4,2,-1,3] => 6 [-4,2,-1,-3] => 7 [-4,-2,1,3] => 6 [-4,-2,1,-3] => 7 [-4,-2,-1,3] => 7 [-4,-2,-1,-3] => 8 [4,2,3,1] => 6 [4,2,3,-1] => 7 [4,2,-3,1] => 7 [4,2,-3,-1] => 8 [4,-2,3,1] => 7 [4,-2,3,-1] => 8 [4,-2,-3,1] => 8 [4,-2,-3,-1] => 9 [-4,2,3,1] => 7 [-4,2,3,-1] => 8 [-4,2,-3,1] => 8 [-4,2,-3,-1] => 9 [-4,-2,3,1] => 8 [-4,-2,3,-1] => 9 [-4,-2,-3,1] => 9 [-4,-2,-3,-1] => 10 [4,3,1,2] => 8 [4,3,1,-2] => 9 [4,3,-1,2] => 9 [4,3,-1,-2] => 10 [4,-3,1,2] => 9 [4,-3,1,-2] => 10 [4,-3,-1,2] => 10 [4,-3,-1,-2] => 11 [-4,3,1,2] => 9 [-4,3,1,-2] => 10 [-4,3,-1,2] => 10 [-4,3,-1,-2] => 11 [-4,-3,1,2] => 10 [-4,-3,1,-2] => 11 [-4,-3,-1,2] => 11 [-4,-3,-1,-2] => 12 [4,3,2,1] => 10 [4,3,2,-1] => 11 [4,3,-2,1] => 11 [4,3,-2,-1] => 12 [4,-3,2,1] => 11 [4,-3,2,-1] => 12 [4,-3,-2,1] => 12 [4,-3,-2,-1] => 13 [-4,3,2,1] => 11 [-4,3,2,-1] => 12 [-4,3,-2,1] => 12 [-4,3,-2,-1] => 13 [-4,-3,2,1] => 12 [-4,-3,2,-1] => 13 [-4,-3,-2,1] => 13 [-4,-3,-2,-1] => 14 ----------------------------------------------------------------------------- Created: Jul 20, 2022 at 08:55 by Dennis Jahn ----------------------------------------------------------------------------- Last Updated: Jul 20, 2022 at 08:55 by Dennis Jahn