***************************************************************************** * 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: St001548 ----------------------------------------------------------------------------- Collection: Decorated permutations ----------------------------------------------------------------------------- Description: The sum of the indices of the first term of the associated Grassmann necklace. Here, we use Postnikov's map (p.59) from decorated permutations to Grassmann necklaces. ----------------------------------------------------------------------------- References: [1] A. Postnikov, ''Total positivity, Grassmannians, and networks.'' 27 Sep 2006. Postnikov, A. Total positivity, Grassmannians, and networks [[arXiv:math/0609764]] ----------------------------------------------------------------------------- Code: ----------------------------------------------------------------------------- Statistic values: [+,+] => 0 [-,+] => 1 [+,-] => 2 [-,-] => 3 [2,1] => 1 [+,+,+] => 0 [-,+,+] => 1 [+,-,+] => 2 [+,+,-] => 3 [-,-,+] => 3 [-,+,-] => 4 [+,-,-] => 5 [-,-,-] => 6 [+,3,2] => 2 [-,3,2] => 3 [2,1,+] => 1 [2,1,-] => 4 [2,3,1] => 1 [3,1,2] => 3 [3,+,1] => 1 [3,-,1] => 3 [+,+,+,+] => 0 [-,+,+,+] => 1 [+,-,+,+] => 2 [+,+,-,+] => 3 [+,+,+,-] => 4 [-,-,+,+] => 3 [-,+,-,+] => 4 [-,+,+,-] => 5 [+,-,-,+] => 5 [+,-,+,-] => 6 [+,+,-,-] => 7 [-,-,-,+] => 6 [-,-,+,-] => 7 [-,+,-,-] => 8 [+,-,-,-] => 9 [-,-,-,-] => 10 [+,+,4,3] => 3 [-,+,4,3] => 4 [+,-,4,3] => 5 [-,-,4,3] => 6 [+,3,2,+] => 2 [-,3,2,+] => 3 [+,3,2,-] => 6 [-,3,2,-] => 7 [+,3,4,2] => 2 [-,3,4,2] => 3 [+,4,2,3] => 5 [-,4,2,3] => 6 [+,4,+,2] => 2 [-,4,+,2] => 3 [+,4,-,2] => 5 [-,4,-,2] => 6 [2,1,+,+] => 1 [2,1,-,+] => 4 [2,1,+,-] => 5 [2,1,-,-] => 8 [2,1,4,3] => 4 [2,3,1,+] => 1 [2,3,1,-] => 5 [2,3,4,1] => 1 [2,4,1,3] => 4 [2,4,+,1] => 1 [2,4,-,1] => 4 [3,1,2,+] => 3 [3,1,2,-] => 7 [3,1,4,2] => 3 [3,+,1,+] => 1 [3,-,1,+] => 3 [3,+,1,-] => 5 [3,-,1,-] => 7 [3,+,4,1] => 1 [3,-,4,1] => 3 [3,4,1,2] => 3 [3,4,2,1] => 3 [4,1,2,3] => 6 [4,1,+,2] => 3 [4,1,-,2] => 6 [4,+,1,3] => 4 [4,-,1,3] => 6 [4,+,+,1] => 1 [4,-,+,1] => 3 [4,+,-,1] => 4 [4,-,-,1] => 6 [4,3,1,2] => 3 [4,3,2,1] => 3 [+,+,+,+,+] => 0 [-,+,+,+,+] => 1 [+,-,+,+,+] => 2 [+,+,-,+,+] => 3 [+,+,+,-,+] => 4 [+,+,+,+,-] => 5 [-,-,+,+,+] => 3 [-,+,-,+,+] => 4 [-,+,+,-,+] => 5 [-,+,+,+,-] => 6 [+,-,-,+,+] => 5 [+,-,+,-,+] => 6 [+,-,+,+,-] => 7 [+,+,-,-,+] => 7 [+,+,-,+,-] => 8 [+,+,+,-,-] => 9 [-,-,-,+,+] => 6 [-,-,+,-,+] => 7 [-,-,+,+,-] => 8 [-,+,-,-,+] => 8 [-,+,-,+,-] => 9 [-,+,+,-,-] => 10 [+,-,-,-,+] => 9 [+,-,-,+,-] => 10 [+,-,+,-,-] => 11 [+,+,-,-,-] => 12 [-,-,-,-,+] => 10 [-,-,-,+,-] => 11 [-,-,+,-,-] => 12 [-,+,-,-,-] => 13 [+,-,-,-,-] => 14 [-,-,-,-,-] => 15 [+,+,+,5,4] => 4 [-,+,+,5,4] => 5 [+,-,+,5,4] => 6 [+,+,-,5,4] => 7 [-,-,+,5,4] => 7 [-,+,-,5,4] => 8 [+,-,-,5,4] => 9 [-,-,-,5,4] => 10 [+,+,4,3,+] => 3 [-,+,4,3,+] => 4 [+,-,4,3,+] => 5 [+,+,4,3,-] => 8 [-,-,4,3,+] => 6 [-,+,4,3,-] => 9 [+,-,4,3,-] => 10 [-,-,4,3,-] => 11 [+,+,4,5,3] => 3 [-,+,4,5,3] => 4 [+,-,4,5,3] => 5 [-,-,4,5,3] => 6 [+,+,5,3,4] => 7 [-,+,5,3,4] => 8 [+,-,5,3,4] => 9 [-,-,5,3,4] => 10 [+,+,5,+,3] => 3 [-,+,5,+,3] => 4 [+,-,5,+,3] => 5 [+,+,5,-,3] => 7 [-,-,5,+,3] => 6 [-,+,5,-,3] => 8 [+,-,5,-,3] => 9 [-,-,5,-,3] => 10 [+,3,2,+,+] => 2 [-,3,2,+,+] => 3 [+,3,2,-,+] => 6 [+,3,2,+,-] => 7 [-,3,2,-,+] => 7 [-,3,2,+,-] => 8 [+,3,2,-,-] => 11 [-,3,2,-,-] => 12 [+,3,2,5,4] => 6 [-,3,2,5,4] => 7 [+,3,4,2,+] => 2 [-,3,4,2,+] => 3 [+,3,4,2,-] => 7 [-,3,4,2,-] => 8 [+,3,4,5,2] => 2 [-,3,4,5,2] => 3 [+,3,5,2,4] => 6 [-,3,5,2,4] => 7 [+,3,5,+,2] => 2 [-,3,5,+,2] => 3 [+,3,5,-,2] => 6 [-,3,5,-,2] => 7 [+,4,2,3,+] => 5 [-,4,2,3,+] => 6 [+,4,2,3,-] => 10 [-,4,2,3,-] => 11 [+,4,2,5,3] => 5 [-,4,2,5,3] => 6 [+,4,+,2,+] => 2 [-,4,+,2,+] => 3 [+,4,-,2,+] => 5 [+,4,+,2,-] => 7 [-,4,-,2,+] => 6 [-,4,+,2,-] => 8 [+,4,-,2,-] => 10 [-,4,-,2,-] => 11 [+,4,+,5,2] => 2 [-,4,+,5,2] => 3 [+,4,-,5,2] => 5 [-,4,-,5,2] => 6 [+,4,5,2,3] => 5 [-,4,5,2,3] => 6 [+,4,5,3,2] => 5 [-,4,5,3,2] => 6 [+,5,2,3,4] => 9 [-,5,2,3,4] => 10 [+,5,2,+,3] => 5 [-,5,2,+,3] => 6 [+,5,2,-,3] => 9 [-,5,2,-,3] => 10 [+,5,+,2,4] => 6 [-,5,+,2,4] => 7 [+,5,-,2,4] => 9 [-,5,-,2,4] => 10 [+,5,+,+,2] => 2 [-,5,+,+,2] => 3 [+,5,-,+,2] => 5 [+,5,+,-,2] => 6 [-,5,-,+,2] => 6 [-,5,+,-,2] => 7 [+,5,-,-,2] => 9 [-,5,-,-,2] => 10 [+,5,4,2,3] => 5 [-,5,4,2,3] => 6 [+,5,4,3,2] => 5 [-,5,4,3,2] => 6 [2,1,+,+,+] => 1 [2,1,-,+,+] => 4 [2,1,+,-,+] => 5 [2,1,+,+,-] => 6 [2,1,-,-,+] => 8 [2,1,-,+,-] => 9 [2,1,+,-,-] => 10 [2,1,-,-,-] => 13 [2,1,+,5,4] => 5 [2,1,-,5,4] => 8 [2,1,4,3,+] => 4 [2,1,4,3,-] => 9 [2,1,4,5,3] => 4 [2,1,5,3,4] => 8 [2,1,5,+,3] => 4 [2,1,5,-,3] => 8 [2,3,1,+,+] => 1 [2,3,1,-,+] => 5 [2,3,1,+,-] => 6 [2,3,1,-,-] => 10 [2,3,1,5,4] => 5 [2,3,4,1,+] => 1 [2,3,4,1,-] => 6 [2,3,4,5,1] => 1 [2,3,5,1,4] => 5 [2,3,5,+,1] => 1 [2,3,5,-,1] => 5 [2,4,1,3,+] => 4 [2,4,1,3,-] => 9 [2,4,1,5,3] => 4 [2,4,+,1,+] => 1 [2,4,-,1,+] => 4 [2,4,+,1,-] => 6 [2,4,-,1,-] => 9 [2,4,+,5,1] => 1 [2,4,-,5,1] => 4 [2,4,5,1,3] => 4 [2,4,5,3,1] => 4 [2,5,1,3,4] => 8 [2,5,1,+,3] => 4 [2,5,1,-,3] => 8 [2,5,+,1,4] => 5 [2,5,-,1,4] => 8 [2,5,+,+,1] => 1 [2,5,-,+,1] => 4 [2,5,+,-,1] => 5 [2,5,-,-,1] => 8 [2,5,4,1,3] => 4 [2,5,4,3,1] => 4 [3,1,2,+,+] => 3 [3,1,2,-,+] => 7 [3,1,2,+,-] => 8 [3,1,2,-,-] => 12 [3,1,2,5,4] => 7 [3,1,4,2,+] => 3 [3,1,4,2,-] => 8 [3,1,4,5,2] => 3 [3,1,5,2,4] => 7 [3,1,5,+,2] => 3 [3,1,5,-,2] => 7 [3,+,1,+,+] => 1 [3,-,1,+,+] => 3 [3,+,1,-,+] => 5 [3,+,1,+,-] => 6 [3,-,1,-,+] => 7 [3,-,1,+,-] => 8 [3,+,1,-,-] => 10 [3,-,1,-,-] => 12 [3,+,1,5,4] => 5 [3,-,1,5,4] => 7 [3,+,4,1,+] => 1 [3,-,4,1,+] => 3 [3,+,4,1,-] => 6 [3,-,4,1,-] => 8 [3,+,4,5,1] => 1 [3,-,4,5,1] => 3 [3,+,5,1,4] => 5 [3,-,5,1,4] => 7 [3,+,5,+,1] => 1 [3,-,5,+,1] => 3 [3,+,5,-,1] => 5 [3,-,5,-,1] => 7 [3,4,1,2,+] => 3 [3,4,1,2,-] => 8 [3,4,1,5,2] => 3 [3,4,2,1,+] => 3 [3,4,2,1,-] => 8 [3,4,2,5,1] => 3 [3,4,5,1,2] => 3 [3,4,5,2,1] => 3 [3,5,1,2,4] => 7 [3,5,1,+,2] => 3 [3,5,1,-,2] => 7 [3,5,2,1,4] => 7 [3,5,2,+,1] => 3 [3,5,2,-,1] => 7 [3,5,4,1,2] => 3 [3,5,4,2,1] => 3 [4,1,2,3,+] => 6 [4,1,2,3,-] => 11 [4,1,2,5,3] => 6 [4,1,+,2,+] => 3 [4,1,-,2,+] => 6 [4,1,+,2,-] => 8 [4,1,-,2,-] => 11 [4,1,+,5,2] => 3 [4,1,-,5,2] => 6 [4,1,5,2,3] => 6 [4,1,5,3,2] => 6 [4,+,1,3,+] => 4 [4,-,1,3,+] => 6 [4,+,1,3,-] => 9 [4,-,1,3,-] => 11 [4,+,1,5,3] => 4 [4,-,1,5,3] => 6 [4,+,+,1,+] => 1 [4,-,+,1,+] => 3 [4,+,-,1,+] => 4 [4,+,+,1,-] => 6 [4,-,-,1,+] => 6 [4,-,+,1,-] => 8 [4,+,-,1,-] => 9 [4,-,-,1,-] => 11 [4,+,+,5,1] => 1 [4,-,+,5,1] => 3 [4,+,-,5,1] => 4 [4,-,-,5,1] => 6 [4,+,5,1,3] => 4 [4,-,5,1,3] => 6 [4,+,5,3,1] => 4 [4,-,5,3,1] => 6 [4,3,1,2,+] => 3 [4,3,1,2,-] => 8 [4,3,1,5,2] => 3 [4,3,2,1,+] => 3 [4,3,2,1,-] => 8 [4,3,2,5,1] => 3 [4,3,5,1,2] => 3 [4,3,5,2,1] => 3 [4,5,1,2,3] => 6 [4,5,1,3,2] => 6 [4,5,2,1,3] => 6 [4,5,2,3,1] => 6 [4,5,+,1,2] => 3 [4,5,-,1,2] => 6 [4,5,+,2,1] => 3 [4,5,-,2,1] => 6 [5,1,2,3,4] => 10 [5,1,2,+,3] => 6 [5,1,2,-,3] => 10 [5,1,+,2,4] => 7 [5,1,-,2,4] => 10 [5,1,+,+,2] => 3 [5,1,-,+,2] => 6 [5,1,+,-,2] => 7 [5,1,-,-,2] => 10 [5,1,4,2,3] => 6 [5,1,4,3,2] => 6 [5,+,1,3,4] => 8 [5,-,1,3,4] => 10 [5,+,1,+,3] => 4 [5,-,1,+,3] => 6 [5,+,1,-,3] => 8 [5,-,1,-,3] => 10 [5,+,+,1,4] => 5 [5,-,+,1,4] => 7 [5,+,-,1,4] => 8 [5,-,-,1,4] => 10 [5,+,+,+,1] => 1 [5,-,+,+,1] => 3 [5,+,-,+,1] => 4 [5,+,+,-,1] => 5 [5,-,-,+,1] => 6 [5,-,+,-,1] => 7 [5,+,-,-,1] => 8 [5,-,-,-,1] => 10 [5,+,4,1,3] => 4 [5,-,4,1,3] => 6 [5,+,4,3,1] => 4 [5,-,4,3,1] => 6 [5,3,1,2,4] => 7 [5,3,1,+,2] => 3 [5,3,1,-,2] => 7 [5,3,2,1,4] => 7 [5,3,2,+,1] => 3 [5,3,2,-,1] => 7 [5,3,4,1,2] => 3 [5,3,4,2,1] => 3 [5,4,1,2,3] => 6 [5,4,1,3,2] => 6 [5,4,2,1,3] => 6 [5,4,2,3,1] => 6 [5,4,+,1,2] => 3 [5,4,-,1,2] => 6 [5,4,+,2,1] => 3 [5,4,-,2,1] => 6 ----------------------------------------------------------------------------- Created: May 14, 2020 at 20:46 by Danny Luecke ----------------------------------------------------------------------------- Last Updated: May 14, 2020 at 22:05 by Danny Luecke