***************************************************************************** * 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: St001544 ----------------------------------------------------------------------------- Collection: Decorated permutations ----------------------------------------------------------------------------- Description: The inversion number of the associated bounded affine permutation. The inversion number is given by $\ell(f) = |\{(i, j) \in [n] \times \mathbb Z \; | \; i < j \text{ and } f(i) > f(j)\}|$ ----------------------------------------------------------------------------- References: [1] Lam, T. Totally nonnegative Grassmannian and Grassmann polytopes [[MathSciNet:3468251]] [[arXiv:1506.00603]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: [+,+] => 0 [-,+] => 0 [+,-] => 1 [-,-] => 0 [2,1] => 0 [+,+,+] => 0 [-,+,+] => 0 [+,-,+] => 1 [+,+,-] => 2 [-,-,+] => 0 [-,+,-] => 1 [+,-,-] => 2 [-,-,-] => 0 [+,3,2] => 1 [-,3,2] => 0 [2,1,+] => 0 [2,1,-] => 1 [2,3,1] => 0 [3,1,2] => 0 [3,+,1] => 1 [3,-,1] => 1 [+,+,+,+] => 0 [-,+,+,+] => 0 [+,-,+,+] => 1 [+,+,-,+] => 2 [+,+,+,-] => 3 [-,-,+,+] => 0 [-,+,-,+] => 1 [-,+,+,-] => 2 [+,-,-,+] => 2 [+,-,+,-] => 3 [+,+,-,-] => 4 [-,-,-,+] => 0 [-,-,+,-] => 1 [-,+,-,-] => 2 [+,-,-,-] => 3 [-,-,-,-] => 0 [+,+,4,3] => 2 [-,+,4,3] => 1 [+,-,4,3] => 2 [-,-,4,3] => 0 [+,3,2,+] => 1 [-,3,2,+] => 0 [+,3,2,-] => 3 [-,3,2,-] => 1 [+,3,4,2] => 2 [-,3,4,2] => 0 [+,4,2,3] => 1 [-,4,2,3] => 0 [+,4,+,2] => 2 [-,4,+,2] => 1 [+,4,-,2] => 3 [-,4,-,2] => 1 [2,1,+,+] => 0 [2,1,-,+] => 1 [2,1,+,-] => 2 [2,1,-,-] => 2 [2,1,4,3] => 1 [2,3,1,+] => 0 [2,3,1,-] => 1 [2,3,4,1] => 0 [2,4,1,3] => 0 [2,4,+,1] => 1 [2,4,-,1] => 1 [3,1,2,+] => 0 [3,1,2,-] => 2 [3,1,4,2] => 1 [3,+,1,+] => 1 [3,-,1,+] => 1 [3,+,1,-] => 3 [3,-,1,-] => 2 [3,+,4,1] => 2 [3,-,4,1] => 1 [3,4,1,2] => 0 [3,4,2,1] => 1 [4,1,2,3] => 0 [4,1,+,2] => 1 [4,1,-,2] => 2 [4,+,1,3] => 1 [4,-,1,3] => 1 [4,+,+,1] => 2 [4,-,+,1] => 2 [4,+,-,1] => 3 [4,-,-,1] => 2 [4,3,1,2] => 1 [4,3,2,1] => 2 [+,+,+,+,+] => 0 [-,+,+,+,+] => 0 [+,-,+,+,+] => 1 [+,+,-,+,+] => 2 [+,+,+,-,+] => 3 [+,+,+,+,-] => 4 [-,-,+,+,+] => 0 [-,+,-,+,+] => 1 [-,+,+,-,+] => 2 [-,+,+,+,-] => 3 [+,-,-,+,+] => 2 [+,-,+,-,+] => 3 [+,-,+,+,-] => 4 [+,+,-,-,+] => 4 [+,+,-,+,-] => 5 [+,+,+,-,-] => 6 [-,-,-,+,+] => 0 [-,-,+,-,+] => 1 [-,-,+,+,-] => 2 [-,+,-,-,+] => 2 [-,+,-,+,-] => 3 [-,+,+,-,-] => 4 [+,-,-,-,+] => 3 [+,-,-,+,-] => 4 [+,-,+,-,-] => 5 [+,+,-,-,-] => 6 [-,-,-,-,+] => 0 [-,-,-,+,-] => 1 [-,-,+,-,-] => 2 [-,+,-,-,-] => 3 [+,-,-,-,-] => 4 [-,-,-,-,-] => 0 [+,+,+,5,4] => 3 [-,+,+,5,4] => 2 [+,-,+,5,4] => 3 [+,+,-,5,4] => 4 [-,-,+,5,4] => 1 [-,+,-,5,4] => 2 [+,-,-,5,4] => 3 [-,-,-,5,4] => 0 [+,+,4,3,+] => 2 [-,+,4,3,+] => 1 [+,-,4,3,+] => 2 [+,+,4,3,-] => 5 [-,-,4,3,+] => 0 [-,+,4,3,-] => 3 [+,-,4,3,-] => 4 [-,-,4,3,-] => 1 [+,+,4,5,3] => 4 [-,+,4,5,3] => 2 [+,-,4,5,3] => 3 [-,-,4,5,3] => 0 [+,+,5,3,4] => 2 [-,+,5,3,4] => 1 [+,-,5,3,4] => 2 [-,-,5,3,4] => 0 [+,+,5,+,3] => 3 [-,+,5,+,3] => 2 [+,-,5,+,3] => 3 [+,+,5,-,3] => 5 [-,-,5,+,3] => 1 [-,+,5,-,3] => 3 [+,-,5,-,3] => 4 [-,-,5,-,3] => 1 [+,3,2,+,+] => 1 [-,3,2,+,+] => 0 [+,3,2,-,+] => 3 [+,3,2,+,-] => 4 [-,3,2,-,+] => 1 [-,3,2,+,-] => 2 [+,3,2,-,-] => 5 [-,3,2,-,-] => 2 [+,3,2,5,4] => 3 [-,3,2,5,4] => 1 [+,3,4,2,+] => 2 [-,3,4,2,+] => 0 [+,3,4,2,-] => 4 [-,3,4,2,-] => 1 [+,3,4,5,2] => 3 [-,3,4,5,2] => 0 [+,3,5,2,4] => 2 [-,3,5,2,4] => 0 [+,3,5,+,2] => 3 [-,3,5,+,2] => 1 [+,3,5,-,2] => 4 [-,3,5,-,2] => 1 [+,4,2,3,+] => 1 [-,4,2,3,+] => 0 [+,4,2,3,-] => 4 [-,4,2,3,-] => 2 [+,4,2,5,3] => 3 [-,4,2,5,3] => 1 [+,4,+,2,+] => 2 [-,4,+,2,+] => 1 [+,4,-,2,+] => 3 [+,4,+,2,-] => 5 [-,4,-,2,+] => 1 [-,4,+,2,-] => 3 [+,4,-,2,-] => 5 [-,4,-,2,-] => 2 [+,4,+,5,2] => 4 [-,4,+,5,2] => 2 [+,4,-,5,2] => 4 [-,4,-,5,2] => 1 [+,4,5,2,3] => 2 [-,4,5,2,3] => 0 [+,4,5,3,2] => 3 [-,4,5,3,2] => 1 [+,5,2,3,4] => 1 [-,5,2,3,4] => 0 [+,5,2,+,3] => 2 [-,5,2,+,3] => 1 [+,5,2,-,3] => 4 [-,5,2,-,3] => 2 [+,5,+,2,4] => 2 [-,5,+,2,4] => 1 [+,5,-,2,4] => 3 [-,5,-,2,4] => 1 [+,5,+,+,2] => 3 [-,5,+,+,2] => 2 [+,5,-,+,2] => 4 [+,5,+,-,2] => 5 [-,5,-,+,2] => 2 [-,5,+,-,2] => 3 [+,5,-,-,2] => 5 [-,5,-,-,2] => 2 [+,5,4,2,3] => 3 [-,5,4,2,3] => 1 [+,5,4,3,2] => 4 [-,5,4,3,2] => 2 [2,1,+,+,+] => 0 [2,1,-,+,+] => 1 [2,1,+,-,+] => 2 [2,1,+,+,-] => 3 [2,1,-,-,+] => 2 [2,1,-,+,-] => 3 [2,1,+,-,-] => 4 [2,1,-,-,-] => 3 [2,1,+,5,4] => 2 [2,1,-,5,4] => 2 [2,1,4,3,+] => 1 [2,1,4,3,-] => 3 [2,1,4,5,3] => 2 [2,1,5,3,4] => 1 [2,1,5,+,3] => 2 [2,1,5,-,3] => 3 [2,3,1,+,+] => 0 [2,3,1,-,+] => 1 [2,3,1,+,-] => 2 [2,3,1,-,-] => 2 [2,3,1,5,4] => 1 [2,3,4,1,+] => 0 [2,3,4,1,-] => 1 [2,3,4,5,1] => 0 [2,3,5,1,4] => 0 [2,3,5,+,1] => 1 [2,3,5,-,1] => 1 [2,4,1,3,+] => 0 [2,4,1,3,-] => 2 [2,4,1,5,3] => 1 [2,4,+,1,+] => 1 [2,4,-,1,+] => 1 [2,4,+,1,-] => 3 [2,4,-,1,-] => 2 [2,4,+,5,1] => 2 [2,4,-,5,1] => 1 [2,4,5,1,3] => 0 [2,4,5,3,1] => 1 [2,5,1,3,4] => 0 [2,5,1,+,3] => 1 [2,5,1,-,3] => 2 [2,5,+,1,4] => 1 [2,5,-,1,4] => 1 [2,5,+,+,1] => 2 [2,5,-,+,1] => 2 [2,5,+,-,1] => 3 [2,5,-,-,1] => 2 [2,5,4,1,3] => 1 [2,5,4,3,1] => 2 [3,1,2,+,+] => 0 [3,1,2,-,+] => 2 [3,1,2,+,-] => 3 [3,1,2,-,-] => 4 [3,1,2,5,4] => 2 [3,1,4,2,+] => 1 [3,1,4,2,-] => 3 [3,1,4,5,2] => 2 [3,1,5,2,4] => 1 [3,1,5,+,2] => 2 [3,1,5,-,2] => 3 [3,+,1,+,+] => 1 [3,-,1,+,+] => 1 [3,+,1,-,+] => 3 [3,+,1,+,-] => 4 [3,-,1,-,+] => 2 [3,-,1,+,-] => 3 [3,+,1,-,-] => 5 [3,-,1,-,-] => 3 [3,+,1,5,4] => 3 [3,-,1,5,4] => 2 [3,+,4,1,+] => 2 [3,-,4,1,+] => 1 [3,+,4,1,-] => 4 [3,-,4,1,-] => 2 [3,+,4,5,1] => 3 [3,-,4,5,1] => 1 [3,+,5,1,4] => 2 [3,-,5,1,4] => 1 [3,+,5,+,1] => 3 [3,-,5,+,1] => 2 [3,+,5,-,1] => 4 [3,-,5,-,1] => 2 [3,4,1,2,+] => 0 [3,4,1,2,-] => 2 [3,4,1,5,2] => 1 [3,4,2,1,+] => 1 [3,4,2,1,-] => 3 [3,4,2,5,1] => 2 [3,4,5,1,2] => 0 [3,4,5,2,1] => 1 [3,5,1,2,4] => 0 [3,5,1,+,2] => 1 [3,5,1,-,2] => 2 [3,5,2,1,4] => 1 [3,5,2,+,1] => 2 [3,5,2,-,1] => 3 [3,5,4,1,2] => 1 [3,5,4,2,1] => 2 [4,1,2,3,+] => 0 [4,1,2,3,-] => 3 [4,1,2,5,3] => 2 [4,1,+,2,+] => 1 [4,1,-,2,+] => 2 [4,1,+,2,-] => 4 [4,1,-,2,-] => 4 [4,1,+,5,2] => 3 [4,1,-,5,2] => 3 [4,1,5,2,3] => 1 [4,1,5,3,2] => 2 [4,+,1,3,+] => 1 [4,-,1,3,+] => 1 [4,+,1,3,-] => 4 [4,-,1,3,-] => 3 [4,+,1,5,3] => 3 [4,-,1,5,3] => 2 [4,+,+,1,+] => 2 [4,-,+,1,+] => 2 [4,+,-,1,+] => 3 [4,+,+,1,-] => 5 [4,-,-,1,+] => 2 [4,-,+,1,-] => 4 [4,+,-,1,-] => 5 [4,-,-,1,-] => 3 [4,+,+,5,1] => 4 [4,-,+,5,1] => 3 [4,+,-,5,1] => 4 [4,-,-,5,1] => 2 [4,+,5,1,3] => 2 [4,-,5,1,3] => 1 [4,+,5,3,1] => 3 [4,-,5,3,1] => 2 [4,3,1,2,+] => 1 [4,3,1,2,-] => 3 [4,3,1,5,2] => 2 [4,3,2,1,+] => 2 [4,3,2,1,-] => 4 [4,3,2,5,1] => 3 [4,3,5,1,2] => 1 [4,3,5,2,1] => 2 [4,5,1,2,3] => 0 [4,5,1,3,2] => 1 [4,5,2,1,3] => 1 [4,5,2,3,1] => 2 [4,5,+,1,2] => 2 [4,5,-,1,2] => 2 [4,5,+,2,1] => 3 [4,5,-,2,1] => 3 [5,1,2,3,4] => 0 [5,1,2,+,3] => 1 [5,1,2,-,3] => 3 [5,1,+,2,4] => 1 [5,1,-,2,4] => 2 [5,1,+,+,2] => 2 [5,1,-,+,2] => 3 [5,1,+,-,2] => 4 [5,1,-,-,2] => 4 [5,1,4,2,3] => 2 [5,1,4,3,2] => 3 [5,+,1,3,4] => 1 [5,-,1,3,4] => 1 [5,+,1,+,3] => 2 [5,-,1,+,3] => 2 [5,+,1,-,3] => 4 [5,-,1,-,3] => 3 [5,+,+,1,4] => 2 [5,-,+,1,4] => 2 [5,+,-,1,4] => 3 [5,-,-,1,4] => 2 [5,+,+,+,1] => 3 [5,-,+,+,1] => 3 [5,+,-,+,1] => 4 [5,+,+,-,1] => 5 [5,-,-,+,1] => 3 [5,-,+,-,1] => 4 [5,+,-,-,1] => 5 [5,-,-,-,1] => 3 [5,+,4,1,3] => 3 [5,-,4,1,3] => 2 [5,+,4,3,1] => 4 [5,-,4,3,1] => 3 [5,3,1,2,4] => 1 [5,3,1,+,2] => 2 [5,3,1,-,2] => 3 [5,3,2,1,4] => 2 [5,3,2,+,1] => 3 [5,3,2,-,1] => 4 [5,3,4,1,2] => 2 [5,3,4,2,1] => 3 [5,4,1,2,3] => 1 [5,4,1,3,2] => 2 [5,4,2,1,3] => 2 [5,4,2,3,1] => 3 [5,4,+,1,2] => 3 [5,4,-,1,2] => 3 [5,4,+,2,1] => 4 [5,4,-,2,1] => 4 ----------------------------------------------------------------------------- Created: May 12, 2020 at 22:38 by Danny Luecke ----------------------------------------------------------------------------- Last Updated: May 13, 2020 at 15:04 by Danny Luecke