***************************************************************************** * 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: St001567 ----------------------------------------------------------------------------- Collection: Decorated permutations ----------------------------------------------------------------------------- Description: The number of weak exceedances of a decorated permutation. A '''weak exceedance''' of a decorated permutation $\tau$ is a position $i$ such that $i > \tau(i)$ or $i = \tau(i)$ and $\tau(i)$ has positive decoration. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: [+] => 1 [-] => 0 [+,+] => 2 [-,+] => 1 [+,-] => 1 [-,-] => 0 [2,1] => 1 [+,+,+] => 3 [-,+,+] => 2 [+,-,+] => 2 [+,+,-] => 2 [-,-,+] => 1 [-,+,-] => 1 [+,-,-] => 1 [-,-,-] => 0 [+,3,2] => 2 [-,3,2] => 1 [2,1,+] => 2 [2,1,-] => 1 [2,3,1] => 2 [3,1,2] => 1 [3,+,1] => 2 [3,-,1] => 1 [+,+,+,+] => 4 [-,+,+,+] => 3 [+,-,+,+] => 3 [+,+,-,+] => 3 [+,+,+,-] => 3 [-,-,+,+] => 2 [-,+,-,+] => 2 [-,+,+,-] => 2 [+,-,-,+] => 2 [+,-,+,-] => 2 [+,+,-,-] => 2 [-,-,-,+] => 1 [-,-,+,-] => 1 [-,+,-,-] => 1 [+,-,-,-] => 1 [-,-,-,-] => 0 [+,+,4,3] => 3 [-,+,4,3] => 2 [+,-,4,3] => 2 [-,-,4,3] => 1 [+,3,2,+] => 3 [-,3,2,+] => 2 [+,3,2,-] => 2 [-,3,2,-] => 1 [+,3,4,2] => 3 [-,3,4,2] => 2 [+,4,2,3] => 2 [-,4,2,3] => 1 [+,4,+,2] => 3 [-,4,+,2] => 2 [+,4,-,2] => 2 [-,4,-,2] => 1 [2,1,+,+] => 3 [2,1,-,+] => 2 [2,1,+,-] => 2 [2,1,-,-] => 1 [2,1,4,3] => 2 [2,3,1,+] => 3 [2,3,1,-] => 2 [2,3,4,1] => 3 [2,4,1,3] => 2 [2,4,+,1] => 3 [2,4,-,1] => 2 [3,1,2,+] => 2 [3,1,2,-] => 1 [3,1,4,2] => 2 [3,+,1,+] => 3 [3,-,1,+] => 2 [3,+,1,-] => 2 [3,-,1,-] => 1 [3,+,4,1] => 3 [3,-,4,1] => 2 [3,4,1,2] => 2 [3,4,2,1] => 2 [4,1,2,3] => 1 [4,1,+,2] => 2 [4,1,-,2] => 1 [4,+,1,3] => 2 [4,-,1,3] => 1 [4,+,+,1] => 3 [4,-,+,1] => 2 [4,+,-,1] => 2 [4,-,-,1] => 1 [4,3,1,2] => 2 [4,3,2,1] => 2 [+,+,+,+,+] => 5 [-,+,+,+,+] => 4 [+,-,+,+,+] => 4 [+,+,-,+,+] => 4 [+,+,+,-,+] => 4 [+,+,+,+,-] => 4 [-,-,+,+,+] => 3 [-,+,-,+,+] => 3 [-,+,+,-,+] => 3 [-,+,+,+,-] => 3 [+,-,-,+,+] => 3 [+,-,+,-,+] => 3 [+,-,+,+,-] => 3 [+,+,-,-,+] => 3 [+,+,-,+,-] => 3 [+,+,+,-,-] => 3 [-,-,-,+,+] => 2 [-,-,+,-,+] => 2 [-,-,+,+,-] => 2 [-,+,-,-,+] => 2 [-,+,-,+,-] => 2 [-,+,+,-,-] => 2 [+,-,-,-,+] => 2 [+,-,-,+,-] => 2 [+,-,+,-,-] => 2 [+,+,-,-,-] => 2 [-,-,-,-,+] => 1 [-,-,-,+,-] => 1 [-,-,+,-,-] => 1 [-,+,-,-,-] => 1 [+,-,-,-,-] => 1 [-,-,-,-,-] => 0 [+,+,+,5,4] => 4 [-,+,+,5,4] => 3 [+,-,+,5,4] => 3 [+,+,-,5,4] => 3 [-,-,+,5,4] => 2 [-,+,-,5,4] => 2 [+,-,-,5,4] => 2 [-,-,-,5,4] => 1 [+,+,4,3,+] => 4 [-,+,4,3,+] => 3 [+,-,4,3,+] => 3 [+,+,4,3,-] => 3 [-,-,4,3,+] => 2 [-,+,4,3,-] => 2 [+,-,4,3,-] => 2 [-,-,4,3,-] => 1 [+,+,4,5,3] => 4 [-,+,4,5,3] => 3 [+,-,4,5,3] => 3 [-,-,4,5,3] => 2 [+,+,5,3,4] => 3 [-,+,5,3,4] => 2 [+,-,5,3,4] => 2 [-,-,5,3,4] => 1 [+,+,5,+,3] => 4 [-,+,5,+,3] => 3 [+,-,5,+,3] => 3 [+,+,5,-,3] => 3 [-,-,5,+,3] => 2 [-,+,5,-,3] => 2 [+,-,5,-,3] => 2 [-,-,5,-,3] => 1 [+,3,2,+,+] => 4 [-,3,2,+,+] => 3 [+,3,2,-,+] => 3 [+,3,2,+,-] => 3 [-,3,2,-,+] => 2 [-,3,2,+,-] => 2 [+,3,2,-,-] => 2 [-,3,2,-,-] => 1 [+,3,2,5,4] => 3 [-,3,2,5,4] => 2 [+,3,4,2,+] => 4 [-,3,4,2,+] => 3 [+,3,4,2,-] => 3 [-,3,4,2,-] => 2 [+,3,4,5,2] => 4 [-,3,4,5,2] => 3 [+,3,5,2,4] => 3 [-,3,5,2,4] => 2 [+,3,5,+,2] => 4 [-,3,5,+,2] => 3 [+,3,5,-,2] => 3 [-,3,5,-,2] => 2 [+,4,2,3,+] => 3 [-,4,2,3,+] => 2 [+,4,2,3,-] => 2 [-,4,2,3,-] => 1 [+,4,2,5,3] => 3 [-,4,2,5,3] => 2 [+,4,+,2,+] => 4 [-,4,+,2,+] => 3 [+,4,-,2,+] => 3 [+,4,+,2,-] => 3 [-,4,-,2,+] => 2 [-,4,+,2,-] => 2 [+,4,-,2,-] => 2 [-,4,-,2,-] => 1 [+,4,+,5,2] => 4 [-,4,+,5,2] => 3 [+,4,-,5,2] => 3 [-,4,-,5,2] => 2 [+,4,5,2,3] => 3 [-,4,5,2,3] => 2 [+,4,5,3,2] => 3 [-,4,5,3,2] => 2 [+,5,2,3,4] => 2 [-,5,2,3,4] => 1 [+,5,2,+,3] => 3 [-,5,2,+,3] => 2 [+,5,2,-,3] => 2 [-,5,2,-,3] => 1 [+,5,+,2,4] => 3 [-,5,+,2,4] => 2 [+,5,-,2,4] => 2 [-,5,-,2,4] => 1 [+,5,+,+,2] => 4 [-,5,+,+,2] => 3 [+,5,-,+,2] => 3 [+,5,+,-,2] => 3 [-,5,-,+,2] => 2 [-,5,+,-,2] => 2 [+,5,-,-,2] => 2 [-,5,-,-,2] => 1 [+,5,4,2,3] => 3 [-,5,4,2,3] => 2 [+,5,4,3,2] => 3 [-,5,4,3,2] => 2 [2,1,+,+,+] => 4 [2,1,-,+,+] => 3 [2,1,+,-,+] => 3 [2,1,+,+,-] => 3 [2,1,-,-,+] => 2 [2,1,-,+,-] => 2 [2,1,+,-,-] => 2 [2,1,-,-,-] => 1 [2,1,+,5,4] => 3 [2,1,-,5,4] => 2 [2,1,4,3,+] => 3 [2,1,4,3,-] => 2 [2,1,4,5,3] => 3 [2,1,5,3,4] => 2 [2,1,5,+,3] => 3 [2,1,5,-,3] => 2 [2,3,1,+,+] => 4 [2,3,1,-,+] => 3 [2,3,1,+,-] => 3 [2,3,1,-,-] => 2 [2,3,1,5,4] => 3 [2,3,4,1,+] => 4 [2,3,4,1,-] => 3 [2,3,4,5,1] => 4 [2,3,5,1,4] => 3 [2,3,5,+,1] => 4 [2,3,5,-,1] => 3 [2,4,1,3,+] => 3 [2,4,1,3,-] => 2 [2,4,1,5,3] => 3 [2,4,+,1,+] => 4 [2,4,-,1,+] => 3 [2,4,+,1,-] => 3 [2,4,-,1,-] => 2 [2,4,+,5,1] => 4 [2,4,-,5,1] => 3 [2,4,5,1,3] => 3 [2,4,5,3,1] => 3 [2,5,1,3,4] => 2 [2,5,1,+,3] => 3 [2,5,1,-,3] => 2 [2,5,+,1,4] => 3 [2,5,-,1,4] => 2 [2,5,+,+,1] => 4 [2,5,-,+,1] => 3 [2,5,+,-,1] => 3 [2,5,-,-,1] => 2 [2,5,4,1,3] => 3 [2,5,4,3,1] => 3 [3,1,2,+,+] => 3 [3,1,2,-,+] => 2 [3,1,2,+,-] => 2 [3,1,2,-,-] => 1 [3,1,2,5,4] => 2 [3,1,4,2,+] => 3 [3,1,4,2,-] => 2 [3,1,4,5,2] => 3 [3,1,5,2,4] => 2 [3,1,5,+,2] => 3 [3,1,5,-,2] => 2 [3,+,1,+,+] => 4 [3,-,1,+,+] => 3 [3,+,1,-,+] => 3 [3,+,1,+,-] => 3 [3,-,1,-,+] => 2 [3,-,1,+,-] => 2 [3,+,1,-,-] => 2 [3,-,1,-,-] => 1 [3,+,1,5,4] => 3 [3,-,1,5,4] => 2 [3,+,4,1,+] => 4 [3,-,4,1,+] => 3 [3,+,4,1,-] => 3 [3,-,4,1,-] => 2 [3,+,4,5,1] => 4 [3,-,4,5,1] => 3 [3,+,5,1,4] => 3 [3,-,5,1,4] => 2 [3,+,5,+,1] => 4 [3,-,5,+,1] => 3 [3,+,5,-,1] => 3 [3,-,5,-,1] => 2 [3,4,1,2,+] => 3 [3,4,1,2,-] => 2 [3,4,1,5,2] => 3 [3,4,2,1,+] => 3 [3,4,2,1,-] => 2 [3,4,2,5,1] => 3 [3,4,5,1,2] => 3 [3,4,5,2,1] => 3 [3,5,1,2,4] => 2 [3,5,1,+,2] => 3 [3,5,1,-,2] => 2 [3,5,2,1,4] => 2 [3,5,2,+,1] => 3 [3,5,2,-,1] => 2 [3,5,4,1,2] => 3 [3,5,4,2,1] => 3 [4,1,2,3,+] => 2 [4,1,2,3,-] => 1 [4,1,2,5,3] => 2 [4,1,+,2,+] => 3 [4,1,-,2,+] => 2 [4,1,+,2,-] => 2 [4,1,-,2,-] => 1 [4,1,+,5,2] => 3 [4,1,-,5,2] => 2 [4,1,5,2,3] => 2 [4,1,5,3,2] => 2 [4,+,1,3,+] => 3 [4,-,1,3,+] => 2 [4,+,1,3,-] => 2 [4,-,1,3,-] => 1 [4,+,1,5,3] => 3 [4,-,1,5,3] => 2 [4,+,+,1,+] => 4 [4,-,+,1,+] => 3 [4,+,-,1,+] => 3 [4,+,+,1,-] => 3 [4,-,-,1,+] => 2 [4,-,+,1,-] => 2 [4,+,-,1,-] => 2 [4,-,-,1,-] => 1 [4,+,+,5,1] => 4 [4,-,+,5,1] => 3 [4,+,-,5,1] => 3 [4,-,-,5,1] => 2 [4,+,5,1,3] => 3 [4,-,5,1,3] => 2 [4,+,5,3,1] => 3 [4,-,5,3,1] => 2 [4,3,1,2,+] => 3 [4,3,1,2,-] => 2 [4,3,1,5,2] => 3 [4,3,2,1,+] => 3 [4,3,2,1,-] => 2 [4,3,2,5,1] => 3 [4,3,5,1,2] => 3 [4,3,5,2,1] => 3 [4,5,1,2,3] => 2 [4,5,1,3,2] => 2 [4,5,2,1,3] => 2 [4,5,2,3,1] => 2 [4,5,+,1,2] => 3 [4,5,-,1,2] => 2 [4,5,+,2,1] => 3 [4,5,-,2,1] => 2 [5,1,2,3,4] => 1 [5,1,2,+,3] => 2 [5,1,2,-,3] => 1 [5,1,+,2,4] => 2 [5,1,-,2,4] => 1 [5,1,+,+,2] => 3 [5,1,-,+,2] => 2 [5,1,+,-,2] => 2 [5,1,-,-,2] => 1 [5,1,4,2,3] => 2 [5,1,4,3,2] => 2 [5,+,1,3,4] => 2 [5,-,1,3,4] => 1 [5,+,1,+,3] => 3 [5,-,1,+,3] => 2 [5,+,1,-,3] => 2 [5,-,1,-,3] => 1 [5,+,+,1,4] => 3 [5,-,+,1,4] => 2 [5,+,-,1,4] => 2 [5,-,-,1,4] => 1 [5,+,+,+,1] => 4 [5,-,+,+,1] => 3 [5,+,-,+,1] => 3 [5,+,+,-,1] => 3 [5,-,-,+,1] => 2 [5,-,+,-,1] => 2 [5,+,-,-,1] => 2 [5,-,-,-,1] => 1 [5,+,4,1,3] => 3 [5,-,4,1,3] => 2 [5,+,4,3,1] => 3 [5,-,4,3,1] => 2 [5,3,1,2,4] => 2 [5,3,1,+,2] => 3 [5,3,1,-,2] => 2 [5,3,2,1,4] => 2 [5,3,2,+,1] => 3 [5,3,2,-,1] => 2 [5,3,4,1,2] => 3 [5,3,4,2,1] => 3 [5,4,1,2,3] => 2 [5,4,1,3,2] => 2 [5,4,2,1,3] => 2 [5,4,2,3,1] => 2 [5,4,+,1,2] => 3 [5,4,-,1,2] => 2 [5,4,+,2,1] => 3 [5,4,-,2,1] => 2 ----------------------------------------------------------------------------- Created: Jul 15, 2020 at 15:35 by Dylan Heuer ----------------------------------------------------------------------------- Last Updated: Jul 15, 2020 at 15:35 by Dylan Heuer