***************************************************************************** * 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: St000333 ----------------------------------------------------------------------------- Collection: Permutations ----------------------------------------------------------------------------- Description: The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. This descent set is denoted by $\operatorname{ZDer}(\sigma)$ in [1]. ----------------------------------------------------------------------------- References: [1] Foata, D., Han, G.-N. Fix-Mahonian Calculus, II: further statistics [[arXiv:math/0703101]] [2] Han, G.-N., Xin, G. Permutations with Extremal number of Fixed Points [[arXiv:0706.1738]] ----------------------------------------------------------------------------- Code: def statistic(x): Z = sz(x) d = 0 for i in range(len(Z)-1): if Z[i] > Z[i+1]: d += 1 return d #Returns the word (as a list) where fixed points of perm are changed to zero (see [2]) def sz(x): l = list(x) for i in x.fixed_points(): l[i-1] = 0 return l ----------------------------------------------------------------------------- Statistic values: [1] => 0 [1,2] => 0 [2,1] => 1 [1,2,3] => 0 [1,3,2] => 1 [2,1,3] => 2 [2,3,1] => 1 [3,1,2] => 1 [3,2,1] => 1 [1,2,3,4] => 0 [1,2,4,3] => 1 [1,3,2,4] => 2 [1,3,4,2] => 1 [1,4,2,3] => 1 [1,4,3,2] => 1 [2,1,3,4] => 2 [2,1,4,3] => 2 [2,3,1,4] => 2 [2,3,4,1] => 1 [2,4,1,3] => 1 [2,4,3,1] => 1 [3,1,2,4] => 2 [3,1,4,2] => 2 [3,2,1,4] => 2 [3,2,4,1] => 2 [3,4,1,2] => 1 [3,4,2,1] => 2 [4,1,2,3] => 1 [4,1,3,2] => 2 [4,2,1,3] => 1 [4,2,3,1] => 1 [4,3,1,2] => 2 [4,3,2,1] => 3 [1,2,3,4,5] => 0 [1,2,3,5,4] => 1 [1,2,4,3,5] => 2 [1,2,4,5,3] => 1 [1,2,5,3,4] => 1 [1,2,5,4,3] => 1 [1,3,2,4,5] => 2 [1,3,2,5,4] => 2 [1,3,4,2,5] => 2 [1,3,4,5,2] => 1 [1,3,5,2,4] => 1 [1,3,5,4,2] => 1 [1,4,2,3,5] => 2 [1,4,2,5,3] => 2 [1,4,3,2,5] => 2 [1,4,3,5,2] => 2 [1,4,5,2,3] => 1 [1,4,5,3,2] => 2 [1,5,2,3,4] => 1 [1,5,2,4,3] => 2 [1,5,3,2,4] => 1 [1,5,3,4,2] => 1 [1,5,4,2,3] => 2 [1,5,4,3,2] => 3 [2,1,3,4,5] => 2 [2,1,3,5,4] => 3 [2,1,4,3,5] => 3 [2,1,4,5,3] => 2 [2,1,5,3,4] => 2 [2,1,5,4,3] => 2 [2,3,1,4,5] => 2 [2,3,1,5,4] => 2 [2,3,4,1,5] => 2 [2,3,4,5,1] => 1 [2,3,5,1,4] => 1 [2,3,5,4,1] => 1 [2,4,1,3,5] => 2 [2,4,1,5,3] => 2 [2,4,3,1,5] => 2 [2,4,3,5,1] => 2 [2,4,5,1,3] => 1 [2,4,5,3,1] => 2 [2,5,1,3,4] => 1 [2,5,1,4,3] => 2 [2,5,3,1,4] => 1 [2,5,3,4,1] => 1 [2,5,4,1,3] => 2 [2,5,4,3,1] => 3 [3,1,2,4,5] => 2 [3,1,2,5,4] => 2 [3,1,4,2,5] => 3 [3,1,4,5,2] => 2 [3,1,5,2,4] => 2 [3,1,5,4,2] => 2 [3,2,1,4,5] => 2 [3,2,1,5,4] => 2 [3,2,4,1,5] => 3 [3,2,4,5,1] => 2 [3,2,5,1,4] => 2 [3,2,5,4,1] => 2 [3,4,1,2,5] => 2 [3,4,1,5,2] => 2 [3,4,2,1,5] => 3 [3,4,2,5,1] => 2 [3,4,5,1,2] => 1 [3,4,5,2,1] => 2 [3,5,1,2,4] => 1 [3,5,1,4,2] => 2 [3,5,2,1,4] => 2 [3,5,2,4,1] => 2 [3,5,4,1,2] => 2 [3,5,4,2,1] => 3 [4,1,2,3,5] => 2 [4,1,2,5,3] => 2 [4,1,3,2,5] => 3 [4,1,3,5,2] => 3 [4,1,5,2,3] => 2 [4,1,5,3,2] => 3 [4,2,1,3,5] => 2 [4,2,1,5,3] => 2 [4,2,3,1,5] => 2 [4,2,3,5,1] => 2 [4,2,5,1,3] => 2 [4,2,5,3,1] => 3 [4,3,1,2,5] => 3 [4,3,1,5,2] => 3 [4,3,2,1,5] => 4 [4,3,2,5,1] => 3 [4,3,5,1,2] => 2 [4,3,5,2,1] => 3 [4,5,1,2,3] => 1 [4,5,1,3,2] => 2 [4,5,2,1,3] => 2 [4,5,2,3,1] => 2 [4,5,3,1,2] => 1 [4,5,3,2,1] => 2 [5,1,2,3,4] => 1 [5,1,2,4,3] => 2 [5,1,3,2,4] => 2 [5,1,3,4,2] => 2 [5,1,4,2,3] => 2 [5,1,4,3,2] => 3 [5,2,1,3,4] => 1 [5,2,1,4,3] => 2 [5,2,3,1,4] => 1 [5,2,3,4,1] => 1 [5,2,4,1,3] => 2 [5,2,4,3,1] => 3 [5,3,1,2,4] => 2 [5,3,1,4,2] => 3 [5,3,2,1,4] => 3 [5,3,2,4,1] => 3 [5,3,4,1,2] => 2 [5,3,4,2,1] => 3 [5,4,1,2,3] => 2 [5,4,1,3,2] => 3 [5,4,2,1,3] => 3 [5,4,2,3,1] => 3 [5,4,3,1,2] => 2 [5,4,3,2,1] => 3 [1,2,3,4,5,6] => 0 [1,2,3,4,6,5] => 1 [1,2,3,5,4,6] => 2 [1,2,3,5,6,4] => 1 [1,2,3,6,4,5] => 1 [1,2,3,6,5,4] => 1 [1,2,4,3,5,6] => 2 [1,2,4,3,6,5] => 2 [1,2,4,5,3,6] => 2 [1,2,4,5,6,3] => 1 [1,2,4,6,3,5] => 1 [1,2,4,6,5,3] => 1 [1,2,5,3,4,6] => 2 [1,2,5,3,6,4] => 2 [1,2,5,4,3,6] => 2 [1,2,5,4,6,3] => 2 [1,2,5,6,3,4] => 1 [1,2,5,6,4,3] => 2 [1,2,6,3,4,5] => 1 [1,2,6,3,5,4] => 2 [1,2,6,4,3,5] => 1 [1,2,6,4,5,3] => 1 [1,2,6,5,3,4] => 2 [1,2,6,5,4,3] => 3 [1,3,2,4,5,6] => 2 [1,3,2,4,6,5] => 3 [1,3,2,5,4,6] => 3 [1,3,2,5,6,4] => 2 [1,3,2,6,4,5] => 2 [1,3,2,6,5,4] => 2 [1,3,4,2,5,6] => 2 [1,3,4,2,6,5] => 2 [1,3,4,5,2,6] => 2 [1,3,4,5,6,2] => 1 [1,3,4,6,2,5] => 1 [1,3,4,6,5,2] => 1 [1,3,5,2,4,6] => 2 [1,3,5,2,6,4] => 2 [1,3,5,4,2,6] => 2 [1,3,5,4,6,2] => 2 [1,3,5,6,2,4] => 1 [1,3,5,6,4,2] => 2 [1,3,6,2,4,5] => 1 [1,3,6,2,5,4] => 2 [1,3,6,4,2,5] => 1 [1,3,6,4,5,2] => 1 [1,3,6,5,2,4] => 2 [1,3,6,5,4,2] => 3 [1,4,2,3,5,6] => 2 [1,4,2,3,6,5] => 2 [1,4,2,5,3,6] => 3 [1,4,2,5,6,3] => 2 [1,4,2,6,3,5] => 2 [1,4,2,6,5,3] => 2 [1,4,3,2,5,6] => 2 [1,4,3,2,6,5] => 2 [1,4,3,5,2,6] => 3 [1,4,3,5,6,2] => 2 [1,4,3,6,2,5] => 2 [1,4,3,6,5,2] => 2 [1,4,5,2,3,6] => 2 [1,4,5,2,6,3] => 2 [1,4,5,3,2,6] => 3 [1,4,5,3,6,2] => 2 [1,4,5,6,2,3] => 1 [1,4,5,6,3,2] => 2 [1,4,6,2,3,5] => 1 [1,4,6,2,5,3] => 2 [1,4,6,3,2,5] => 2 [1,4,6,3,5,2] => 2 [1,4,6,5,2,3] => 2 [1,4,6,5,3,2] => 3 [1,5,2,3,4,6] => 2 [1,5,2,3,6,4] => 2 [1,5,2,4,3,6] => 3 [1,5,2,4,6,3] => 3 [1,5,2,6,3,4] => 2 [1,5,2,6,4,3] => 3 [1,5,3,2,4,6] => 2 [1,5,3,2,6,4] => 2 [1,5,3,4,2,6] => 2 [1,5,3,4,6,2] => 2 [1,5,3,6,2,4] => 2 [1,5,3,6,4,2] => 3 [1,5,4,2,3,6] => 3 [1,5,4,2,6,3] => 3 [1,5,4,3,2,6] => 4 [1,5,4,3,6,2] => 3 [1,5,4,6,2,3] => 2 [1,5,4,6,3,2] => 3 [1,5,6,2,3,4] => 1 [1,5,6,2,4,3] => 2 [1,5,6,3,2,4] => 2 [1,5,6,3,4,2] => 2 [1,5,6,4,2,3] => 1 [1,5,6,4,3,2] => 2 [1,6,2,3,4,5] => 1 [1,6,2,3,5,4] => 2 [1,6,2,4,3,5] => 2 [1,6,2,4,5,3] => 2 [1,6,2,5,3,4] => 2 [1,6,2,5,4,3] => 3 [1,6,3,2,4,5] => 1 [1,6,3,2,5,4] => 2 [1,6,3,4,2,5] => 1 [1,6,3,4,5,2] => 1 [1,6,3,5,2,4] => 2 [1,6,3,5,4,2] => 3 [1,6,4,2,3,5] => 2 [1,6,4,2,5,3] => 3 [1,6,4,3,2,5] => 3 [1,6,4,3,5,2] => 3 [1,6,4,5,2,3] => 2 [1,6,4,5,3,2] => 3 [1,6,5,2,3,4] => 2 [1,6,5,2,4,3] => 3 [1,6,5,3,2,4] => 3 [1,6,5,3,4,2] => 3 [1,6,5,4,2,3] => 2 [1,6,5,4,3,2] => 3 [2,1,3,4,5,6] => 2 [2,1,3,4,6,5] => 3 [2,1,3,5,4,6] => 4 [2,1,3,5,6,4] => 3 [2,1,3,6,4,5] => 3 [2,1,3,6,5,4] => 3 [2,1,4,3,5,6] => 3 [2,1,4,3,6,5] => 3 [2,1,4,5,3,6] => 3 [2,1,4,5,6,3] => 2 [2,1,4,6,3,5] => 2 [2,1,4,6,5,3] => 2 [2,1,5,3,4,6] => 3 [2,1,5,3,6,4] => 3 [2,1,5,4,3,6] => 3 [2,1,5,4,6,3] => 3 [2,1,5,6,3,4] => 2 [2,1,5,6,4,3] => 3 [2,1,6,3,4,5] => 2 [2,1,6,3,5,4] => 3 [2,1,6,4,3,5] => 2 [2,1,6,4,5,3] => 2 [2,1,6,5,3,4] => 3 [2,1,6,5,4,3] => 4 [2,3,1,4,5,6] => 2 [2,3,1,4,6,5] => 3 [2,3,1,5,4,6] => 3 [2,3,1,5,6,4] => 2 [2,3,1,6,4,5] => 2 [2,3,1,6,5,4] => 2 [2,3,4,1,5,6] => 2 [2,3,4,1,6,5] => 2 [2,3,4,5,1,6] => 2 [2,3,4,5,6,1] => 1 [2,3,4,6,1,5] => 1 [2,3,4,6,5,1] => 1 [2,3,5,1,4,6] => 2 [2,3,5,1,6,4] => 2 [2,3,5,4,1,6] => 2 [2,3,5,4,6,1] => 2 [2,3,5,6,1,4] => 1 [2,3,5,6,4,1] => 2 [2,3,6,1,4,5] => 1 [2,3,6,1,5,4] => 2 [2,3,6,4,1,5] => 1 [2,3,6,4,5,1] => 1 [2,3,6,5,1,4] => 2 [2,3,6,5,4,1] => 3 [2,4,1,3,5,6] => 2 [2,4,1,3,6,5] => 2 [2,4,1,5,3,6] => 3 [2,4,1,5,6,3] => 2 [2,4,1,6,3,5] => 2 [2,4,1,6,5,3] => 2 [2,4,3,1,5,6] => 2 [2,4,3,1,6,5] => 2 [2,4,3,5,1,6] => 3 [2,4,3,5,6,1] => 2 [2,4,3,6,1,5] => 2 [2,4,3,6,5,1] => 2 [2,4,5,1,3,6] => 2 [2,4,5,1,6,3] => 2 [2,4,5,3,1,6] => 3 [2,4,5,3,6,1] => 2 [2,4,5,6,1,3] => 1 [2,4,5,6,3,1] => 2 [2,4,6,1,3,5] => 1 [2,4,6,1,5,3] => 2 [2,4,6,3,1,5] => 2 [2,4,6,3,5,1] => 2 [2,4,6,5,1,3] => 2 [2,4,6,5,3,1] => 3 [2,5,1,3,4,6] => 2 [2,5,1,3,6,4] => 2 [2,5,1,4,3,6] => 3 [2,5,1,4,6,3] => 3 [2,5,1,6,3,4] => 2 [2,5,1,6,4,3] => 3 [2,5,3,1,4,6] => 2 [2,5,3,1,6,4] => 2 [2,5,3,4,1,6] => 2 [2,5,3,4,6,1] => 2 [2,5,3,6,1,4] => 2 [2,5,3,6,4,1] => 3 [2,5,4,1,3,6] => 3 [2,5,4,1,6,3] => 3 [2,5,4,3,1,6] => 4 [2,5,4,3,6,1] => 3 [2,5,4,6,1,3] => 2 [2,5,4,6,3,1] => 3 [2,5,6,1,3,4] => 1 [2,5,6,1,4,3] => 2 [2,5,6,3,1,4] => 2 [2,5,6,3,4,1] => 2 [2,5,6,4,1,3] => 1 [2,5,6,4,3,1] => 2 [2,6,1,3,4,5] => 1 [2,6,1,3,5,4] => 2 [2,6,1,4,3,5] => 2 [2,6,1,4,5,3] => 2 [2,6,1,5,3,4] => 2 [2,6,1,5,4,3] => 3 [2,6,3,1,4,5] => 1 [2,6,3,1,5,4] => 2 [2,6,3,4,1,5] => 1 [2,6,3,4,5,1] => 1 [2,6,3,5,1,4] => 2 [2,6,3,5,4,1] => 3 [2,6,4,1,3,5] => 2 [2,6,4,1,5,3] => 3 [2,6,4,3,1,5] => 3 [2,6,4,3,5,1] => 3 [2,6,4,5,1,3] => 2 [2,6,4,5,3,1] => 3 [2,6,5,1,3,4] => 2 [2,6,5,1,4,3] => 3 [2,6,5,3,1,4] => 3 [2,6,5,3,4,1] => 3 [2,6,5,4,1,3] => 2 [2,6,5,4,3,1] => 3 [3,1,2,4,5,6] => 2 [3,1,2,4,6,5] => 3 [3,1,2,5,4,6] => 3 [3,1,2,5,6,4] => 2 [3,1,2,6,4,5] => 2 [3,1,2,6,5,4] => 2 [3,1,4,2,5,6] => 3 [3,1,4,2,6,5] => 3 [3,1,4,5,2,6] => 3 [3,1,4,5,6,2] => 2 [3,1,4,6,2,5] => 2 [3,1,4,6,5,2] => 2 [3,1,5,2,4,6] => 3 [3,1,5,2,6,4] => 3 [3,1,5,4,2,6] => 3 [3,1,5,4,6,2] => 3 [3,1,5,6,2,4] => 2 [3,1,5,6,4,2] => 3 [3,1,6,2,4,5] => 2 [3,1,6,2,5,4] => 3 [3,1,6,4,2,5] => 2 [3,1,6,4,5,2] => 2 [3,1,6,5,2,4] => 3 [3,1,6,5,4,2] => 4 [3,2,1,4,5,6] => 2 [3,2,1,4,6,5] => 3 [3,2,1,5,4,6] => 3 [3,2,1,5,6,4] => 2 [3,2,1,6,4,5] => 2 [3,2,1,6,5,4] => 2 [3,2,4,1,5,6] => 3 [3,2,4,1,6,5] => 3 [3,2,4,5,1,6] => 3 [3,2,4,5,6,1] => 2 [3,2,4,6,1,5] => 2 [3,2,4,6,5,1] => 2 [3,2,5,1,4,6] => 3 [3,2,5,1,6,4] => 3 [3,2,5,4,1,6] => 3 [3,2,5,4,6,1] => 3 [3,2,5,6,1,4] => 2 [3,2,5,6,4,1] => 3 [3,2,6,1,4,5] => 2 [3,2,6,1,5,4] => 3 [3,2,6,4,1,5] => 2 [3,2,6,4,5,1] => 2 [3,2,6,5,1,4] => 3 [3,2,6,5,4,1] => 4 [3,4,1,2,5,6] => 2 [3,4,1,2,6,5] => 2 [3,4,1,5,2,6] => 3 [3,4,1,5,6,2] => 2 [3,4,1,6,2,5] => 2 [3,4,1,6,5,2] => 2 [3,4,2,1,5,6] => 3 [3,4,2,1,6,5] => 3 [3,4,2,5,1,6] => 3 [3,4,2,5,6,1] => 2 [3,4,2,6,1,5] => 2 [3,4,2,6,5,1] => 2 [3,4,5,1,2,6] => 2 [3,4,5,1,6,2] => 2 [3,4,5,2,1,6] => 3 [3,4,5,2,6,1] => 2 [3,4,5,6,1,2] => 1 [3,4,5,6,2,1] => 2 [3,4,6,1,2,5] => 1 [3,4,6,1,5,2] => 2 [3,4,6,2,1,5] => 2 [3,4,6,2,5,1] => 2 [3,4,6,5,1,2] => 2 [3,4,6,5,2,1] => 3 [3,5,1,2,4,6] => 2 [3,5,1,2,6,4] => 2 [3,5,1,4,2,6] => 3 [3,5,1,4,6,2] => 3 [3,5,1,6,2,4] => 2 [3,5,1,6,4,2] => 3 [3,5,2,1,4,6] => 3 [3,5,2,1,6,4] => 3 [3,5,2,4,1,6] => 3 [3,5,2,4,6,1] => 3 [3,5,2,6,1,4] => 2 [3,5,2,6,4,1] => 3 [3,5,4,1,2,6] => 3 [3,5,4,1,6,2] => 3 [3,5,4,2,1,6] => 4 [3,5,4,2,6,1] => 3 [3,5,4,6,1,2] => 2 [3,5,4,6,2,1] => 3 [3,5,6,1,2,4] => 1 [3,5,6,1,4,2] => 2 [3,5,6,2,1,4] => 2 [3,5,6,2,4,1] => 2 [3,5,6,4,1,2] => 1 [3,5,6,4,2,1] => 2 [3,6,1,2,4,5] => 1 [3,6,1,2,5,4] => 2 [3,6,1,4,2,5] => 2 [3,6,1,4,5,2] => 2 [3,6,1,5,2,4] => 2 [3,6,1,5,4,2] => 3 [3,6,2,1,4,5] => 2 [3,6,2,1,5,4] => 3 [3,6,2,4,1,5] => 2 [3,6,2,4,5,1] => 2 [3,6,2,5,1,4] => 2 [3,6,2,5,4,1] => 3 [3,6,4,1,2,5] => 2 [3,6,4,1,5,2] => 3 [3,6,4,2,1,5] => 3 [3,6,4,2,5,1] => 3 [3,6,4,5,1,2] => 2 [3,6,4,5,2,1] => 3 [3,6,5,1,2,4] => 2 [3,6,5,1,4,2] => 3 [3,6,5,2,1,4] => 3 [3,6,5,2,4,1] => 3 [3,6,5,4,1,2] => 2 [3,6,5,4,2,1] => 3 [4,1,2,3,5,6] => 2 [4,1,2,3,6,5] => 2 [4,1,2,5,3,6] => 3 [4,1,2,5,6,3] => 2 [4,1,2,6,3,5] => 2 [4,1,2,6,5,3] => 2 [4,1,3,2,5,6] => 3 [4,1,3,2,6,5] => 3 [4,1,3,5,2,6] => 4 [4,1,3,5,6,2] => 3 [4,1,3,6,2,5] => 3 [4,1,3,6,5,2] => 3 [4,1,5,2,3,6] => 3 [4,1,5,2,6,3] => 3 [4,1,5,3,2,6] => 4 [4,1,5,3,6,2] => 3 [4,1,5,6,2,3] => 2 [4,1,5,6,3,2] => 3 [4,1,6,2,3,5] => 2 [4,1,6,2,5,3] => 3 [4,1,6,3,2,5] => 3 [4,1,6,3,5,2] => 3 [4,1,6,5,2,3] => 3 [4,1,6,5,3,2] => 4 [4,2,1,3,5,6] => 2 [4,2,1,3,6,5] => 2 [4,2,1,5,3,6] => 3 [4,2,1,5,6,3] => 2 [4,2,1,6,3,5] => 2 [4,2,1,6,5,3] => 2 [4,2,3,1,5,6] => 2 [4,2,3,1,6,5] => 2 [4,2,3,5,1,6] => 3 [4,2,3,5,6,1] => 2 [4,2,3,6,1,5] => 2 [4,2,3,6,5,1] => 2 [4,2,5,1,3,6] => 3 [4,2,5,1,6,3] => 3 [4,2,5,3,1,6] => 4 [4,2,5,3,6,1] => 3 [4,2,5,6,1,3] => 2 [4,2,5,6,3,1] => 3 [4,2,6,1,3,5] => 2 [4,2,6,1,5,3] => 3 [4,2,6,3,1,5] => 3 [4,2,6,3,5,1] => 3 [4,2,6,5,1,3] => 3 [4,2,6,5,3,1] => 4 [4,3,1,2,5,6] => 3 [4,3,1,2,6,5] => 3 [4,3,1,5,2,6] => 4 [4,3,1,5,6,2] => 3 [4,3,1,6,2,5] => 3 [4,3,1,6,5,2] => 3 [4,3,2,1,5,6] => 4 [4,3,2,1,6,5] => 4 [4,3,2,5,1,6] => 4 [4,3,2,5,6,1] => 3 [4,3,2,6,1,5] => 3 [4,3,2,6,5,1] => 3 [4,3,5,1,2,6] => 3 [4,3,5,1,6,2] => 3 [4,3,5,2,1,6] => 4 [4,3,5,2,6,1] => 3 [4,3,5,6,1,2] => 2 [4,3,5,6,2,1] => 3 [4,3,6,1,2,5] => 2 [4,3,6,1,5,2] => 3 [4,3,6,2,1,5] => 3 [4,3,6,2,5,1] => 3 [4,3,6,5,1,2] => 3 [4,3,6,5,2,1] => 4 [4,5,1,2,3,6] => 2 [4,5,1,2,6,3] => 2 [4,5,1,3,2,6] => 3 [4,5,1,3,6,2] => 2 [4,5,1,6,2,3] => 2 [4,5,1,6,3,2] => 3 [4,5,2,1,3,6] => 3 [4,5,2,1,6,3] => 3 [4,5,2,3,1,6] => 3 [4,5,2,3,6,1] => 2 [4,5,2,6,1,3] => 2 [4,5,2,6,3,1] => 3 [4,5,3,1,2,6] => 2 [4,5,3,1,6,2] => 2 [4,5,3,2,1,6] => 3 [4,5,3,2,6,1] => 2 [4,5,3,6,1,2] => 2 [4,5,3,6,2,1] => 3 [4,5,6,1,2,3] => 1 [4,5,6,1,3,2] => 2 [4,5,6,2,1,3] => 2 [4,5,6,2,3,1] => 2 [4,5,6,3,1,2] => 2 [4,5,6,3,2,1] => 3 [4,6,1,2,3,5] => 1 [4,6,1,2,5,3] => 2 [4,6,1,3,2,5] => 2 [4,6,1,3,5,2] => 2 [4,6,1,5,2,3] => 2 [4,6,1,5,3,2] => 3 [4,6,2,1,3,5] => 2 [4,6,2,1,5,3] => 3 [4,6,2,3,1,5] => 2 [4,6,2,3,5,1] => 2 [4,6,2,5,1,3] => 2 [4,6,2,5,3,1] => 3 [4,6,3,1,2,5] => 1 [4,6,3,1,5,2] => 2 [4,6,3,2,1,5] => 2 [4,6,3,2,5,1] => 2 [4,6,3,5,1,2] => 2 [4,6,3,5,2,1] => 3 [4,6,5,1,2,3] => 2 [4,6,5,1,3,2] => 3 [4,6,5,2,1,3] => 3 [4,6,5,2,3,1] => 3 [4,6,5,3,1,2] => 3 [4,6,5,3,2,1] => 4 [5,1,2,3,4,6] => 2 [5,1,2,3,6,4] => 2 [5,1,2,4,3,6] => 3 [5,1,2,4,6,3] => 3 [5,1,2,6,3,4] => 2 [5,1,2,6,4,3] => 3 [5,1,3,2,4,6] => 3 [5,1,3,2,6,4] => 3 [5,1,3,4,2,6] => 3 [5,1,3,4,6,2] => 3 [5,1,3,6,2,4] => 3 [5,1,3,6,4,2] => 4 [5,1,4,2,3,6] => 3 [5,1,4,2,6,3] => 3 [5,1,4,3,2,6] => 4 [5,1,4,3,6,2] => 3 [5,1,4,6,2,3] => 2 [5,1,4,6,3,2] => 3 [5,1,6,2,3,4] => 2 [5,1,6,2,4,3] => 3 [5,1,6,3,2,4] => 3 [5,1,6,3,4,2] => 3 [5,1,6,4,2,3] => 2 [5,1,6,4,3,2] => 3 [5,2,1,3,4,6] => 2 [5,2,1,3,6,4] => 2 [5,2,1,4,3,6] => 3 [5,2,1,4,6,3] => 3 [5,2,1,6,3,4] => 2 [5,2,1,6,4,3] => 3 [5,2,3,1,4,6] => 2 [5,2,3,1,6,4] => 2 [5,2,3,4,1,6] => 2 [5,2,3,4,6,1] => 2 [5,2,3,6,1,4] => 2 [5,2,3,6,4,1] => 3 [5,2,4,1,3,6] => 3 [5,2,4,1,6,3] => 3 [5,2,4,3,1,6] => 4 [5,2,4,3,6,1] => 3 [5,2,4,6,1,3] => 2 [5,2,4,6,3,1] => 3 [5,2,6,1,3,4] => 2 [5,2,6,1,4,3] => 3 [5,2,6,3,1,4] => 3 [5,2,6,3,4,1] => 3 [5,2,6,4,1,3] => 2 [5,2,6,4,3,1] => 3 [5,3,1,2,4,6] => 3 [5,3,1,2,6,4] => 3 [5,3,1,4,2,6] => 4 [5,3,1,4,6,2] => 4 [5,3,1,6,2,4] => 3 [5,3,1,6,4,2] => 4 [5,3,2,1,4,6] => 4 [5,3,2,1,6,4] => 4 [5,3,2,4,1,6] => 4 [5,3,2,4,6,1] => 4 [5,3,2,6,1,4] => 3 [5,3,2,6,4,1] => 4 [5,3,4,1,2,6] => 3 [5,3,4,1,6,2] => 3 [5,3,4,2,1,6] => 4 [5,3,4,2,6,1] => 3 [5,3,4,6,1,2] => 2 [5,3,4,6,2,1] => 3 [5,3,6,1,2,4] => 2 [5,3,6,1,4,2] => 3 [5,3,6,2,1,4] => 3 [5,3,6,2,4,1] => 3 [5,3,6,4,1,2] => 2 [5,3,6,4,2,1] => 3 [5,4,1,2,3,6] => 3 [5,4,1,2,6,3] => 3 [5,4,1,3,2,6] => 4 [5,4,1,3,6,2] => 3 [5,4,1,6,2,3] => 3 [5,4,1,6,3,2] => 4 [5,4,2,1,3,6] => 4 [5,4,2,1,6,3] => 4 [5,4,2,3,1,6] => 4 [5,4,2,3,6,1] => 3 [5,4,2,6,1,3] => 3 [5,4,2,6,3,1] => 4 [5,4,3,1,2,6] => 3 [5,4,3,1,6,2] => 3 [5,4,3,2,1,6] => 4 [5,4,3,2,6,1] => 3 [5,4,3,6,1,2] => 3 [5,4,3,6,2,1] => 4 [5,4,6,1,2,3] => 2 [5,4,6,1,3,2] => 3 [5,4,6,2,1,3] => 3 [5,4,6,2,3,1] => 3 [5,4,6,3,1,2] => 3 [5,4,6,3,2,1] => 4 [5,6,1,2,3,4] => 1 [5,6,1,2,4,3] => 2 [5,6,1,3,2,4] => 2 [5,6,1,3,4,2] => 2 [5,6,1,4,2,3] => 2 [5,6,1,4,3,2] => 3 [5,6,2,1,3,4] => 2 [5,6,2,1,4,3] => 3 [5,6,2,3,1,4] => 2 [5,6,2,3,4,1] => 2 [5,6,2,4,1,3] => 2 [5,6,2,4,3,1] => 3 [5,6,3,1,2,4] => 1 [5,6,3,1,4,2] => 2 [5,6,3,2,1,4] => 2 [5,6,3,2,4,1] => 2 [5,6,3,4,1,2] => 1 [5,6,3,4,2,1] => 2 [5,6,4,1,2,3] => 2 [5,6,4,1,3,2] => 3 [5,6,4,2,1,3] => 3 [5,6,4,2,3,1] => 3 [5,6,4,3,1,2] => 3 [5,6,4,3,2,1] => 4 [6,1,2,3,4,5] => 1 [6,1,2,3,5,4] => 2 [6,1,2,4,3,5] => 2 [6,1,2,4,5,3] => 2 [6,1,2,5,3,4] => 2 [6,1,2,5,4,3] => 3 [6,1,3,2,4,5] => 2 [6,1,3,2,5,4] => 3 [6,1,3,4,2,5] => 2 [6,1,3,4,5,2] => 2 [6,1,3,5,2,4] => 3 [6,1,3,5,4,2] => 4 [6,1,4,2,3,5] => 2 [6,1,4,2,5,3] => 3 [6,1,4,3,2,5] => 3 [6,1,4,3,5,2] => 3 [6,1,4,5,2,3] => 2 [6,1,4,5,3,2] => 3 [6,1,5,2,3,4] => 2 [6,1,5,2,4,3] => 3 [6,1,5,3,2,4] => 3 [6,1,5,3,4,2] => 3 [6,1,5,4,2,3] => 2 [6,1,5,4,3,2] => 3 [6,2,1,3,4,5] => 1 [6,2,1,3,5,4] => 2 [6,2,1,4,3,5] => 2 [6,2,1,4,5,3] => 2 [6,2,1,5,3,4] => 2 [6,2,1,5,4,3] => 3 [6,2,3,1,4,5] => 1 [6,2,3,1,5,4] => 2 [6,2,3,4,1,5] => 1 [6,2,3,4,5,1] => 1 [6,2,3,5,1,4] => 2 [6,2,3,5,4,1] => 3 [6,2,4,1,3,5] => 2 [6,2,4,1,5,3] => 3 [6,2,4,3,1,5] => 3 [6,2,4,3,5,1] => 3 [6,2,4,5,1,3] => 2 [6,2,4,5,3,1] => 3 [6,2,5,1,3,4] => 2 [6,2,5,1,4,3] => 3 [6,2,5,3,1,4] => 3 [6,2,5,3,4,1] => 3 [6,2,5,4,1,3] => 2 [6,2,5,4,3,1] => 3 [6,3,1,2,4,5] => 2 [6,3,1,2,5,4] => 3 [6,3,1,4,2,5] => 3 [6,3,1,4,5,2] => 3 [6,3,1,5,2,4] => 3 [6,3,1,5,4,2] => 4 [6,3,2,1,4,5] => 3 [6,3,2,1,5,4] => 4 [6,3,2,4,1,5] => 3 [6,3,2,4,5,1] => 3 [6,3,2,5,1,4] => 3 [6,3,2,5,4,1] => 4 [6,3,4,1,2,5] => 2 [6,3,4,1,5,2] => 3 [6,3,4,2,1,5] => 3 [6,3,4,2,5,1] => 3 [6,3,4,5,1,2] => 2 [6,3,4,5,2,1] => 3 [6,3,5,1,2,4] => 2 [6,3,5,1,4,2] => 3 [6,3,5,2,1,4] => 3 [6,3,5,2,4,1] => 3 [6,3,5,4,1,2] => 2 [6,3,5,4,2,1] => 3 [6,4,1,2,3,5] => 2 [6,4,1,2,5,3] => 3 [6,4,1,3,2,5] => 3 [6,4,1,3,5,2] => 3 [6,4,1,5,2,3] => 3 [6,4,1,5,3,2] => 4 [6,4,2,1,3,5] => 3 [6,4,2,1,5,3] => 4 [6,4,2,3,1,5] => 3 [6,4,2,3,5,1] => 3 [6,4,2,5,1,3] => 3 [6,4,2,5,3,1] => 4 [6,4,3,1,2,5] => 2 [6,4,3,1,5,2] => 3 [6,4,3,2,1,5] => 3 [6,4,3,2,5,1] => 3 [6,4,3,5,1,2] => 3 [6,4,3,5,2,1] => 4 [6,4,5,1,2,3] => 2 [6,4,5,1,3,2] => 3 [6,4,5,2,1,3] => 3 [6,4,5,2,3,1] => 3 [6,4,5,3,1,2] => 3 [6,4,5,3,2,1] => 4 [6,5,1,2,3,4] => 2 [6,5,1,2,4,3] => 3 [6,5,1,3,2,4] => 3 [6,5,1,3,4,2] => 3 [6,5,1,4,2,3] => 3 [6,5,1,4,3,2] => 4 [6,5,2,1,3,4] => 3 [6,5,2,1,4,3] => 4 [6,5,2,3,1,4] => 3 [6,5,2,3,4,1] => 3 [6,5,2,4,1,3] => 3 [6,5,2,4,3,1] => 4 [6,5,3,1,2,4] => 2 [6,5,3,1,4,2] => 3 [6,5,3,2,1,4] => 3 [6,5,3,2,4,1] => 3 [6,5,3,4,1,2] => 2 [6,5,3,4,2,1] => 3 [6,5,4,1,2,3] => 3 [6,5,4,1,3,2] => 4 [6,5,4,2,1,3] => 4 [6,5,4,2,3,1] => 4 [6,5,4,3,1,2] => 4 [6,5,4,3,2,1] => 5 ----------------------------------------------------------------------------- Created: Dec 18, 2015 at 13:44 by Joseph Bernstein ----------------------------------------------------------------------------- Last Updated: Apr 07, 2016 at 07:59 by Martin Rubey