***************************************************************************** * 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: St000034 ----------------------------------------------------------------------------- Collection: Permutations ----------------------------------------------------------------------------- Description: The maximum defect over any reduced expression for a permutation and any subexpression. ----------------------------------------------------------------------------- References: [1] Deodhar, V. V. A combinatorial setting for questions in Kazhdan-Lusztig theory [[MathSciNet:1065215]] [2] Billey, S. C., Warrington, G. S. Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations [[MathSciNet:1826948]] [3] Jones, B., Woo, A. Mask formulas for cograssmannian Kazhdan-Lusztig polynomials [[arXiv:1011.1110]] ----------------------------------------------------------------------------- Code: # WARNING: takes a long time even for permutations of 5. def defect_set(sigma, w): """ INPUT: - w is a reduced word for a permutation - sigma is the mask, given as a binary word of length l(w) OUTPUT: - the defect set, that is the set of positions j such that l(w^{\sigma[j-1]} w_j) < l(w^{\sigma[j-1]}) """ res = [] S = Permutations(max(w)+1) w_S = [S.simple_reflection(e) for e in w] w_up_to_j = S([]) for j in range(len(w)-1): if sigma[j] == 1: w_up_to_j = w_up_to_j.left_action_product(w_S[j]) if (w_up_to_j.left_action_product(w_S[j+1])).length() < w_up_to_j.length(): res += [j+2] return res def statistic(pi): """ sage: statistic(Permutation([4,3,5,2,1])) 3 sage: statistic(Permutation([5,4,2,1,3])) 3 """ k = pi.length() if not k: return 0 masks = cartesian_product([[0,1]]*k) return max([len(defect_set(m, w)) for m in masks for w in pi.reduced_words()]) ----------------------------------------------------------------------------- Statistic values: [] => 0 [1] => 0 [1,2] => 0 [2,1] => 0 [1,2,3] => 0 [1,3,2] => 0 [2,1,3] => 0 [2,3,1] => 0 [3,1,2] => 0 [3,2,1] => 1 [1,2,3,4] => 0 [1,2,4,3] => 0 [1,3,2,4] => 0 [1,3,4,2] => 0 [1,4,2,3] => 0 [1,4,3,2] => 1 [2,1,3,4] => 0 [2,1,4,3] => 0 [2,3,1,4] => 0 [2,3,4,1] => 0 [2,4,1,3] => 0 [2,4,3,1] => 1 [3,1,2,4] => 0 [3,1,4,2] => 0 [3,2,1,4] => 1 [3,2,4,1] => 1 [3,4,1,2] => 1 [3,4,2,1] => 2 [4,1,2,3] => 0 [4,1,3,2] => 1 [4,2,1,3] => 1 [4,2,3,1] => 2 [4,3,1,2] => 2 [4,3,2,1] => 2 [1,2,3,4,5] => 0 [1,2,3,5,4] => 0 [1,2,4,3,5] => 0 [1,2,4,5,3] => 0 [1,2,5,3,4] => 0 [1,2,5,4,3] => 1 [1,3,2,4,5] => 0 [1,3,2,5,4] => 0 [1,3,4,2,5] => 0 [1,3,4,5,2] => 0 [1,3,5,2,4] => 0 [1,3,5,4,2] => 1 [1,4,2,3,5] => 0 [1,4,2,5,3] => 0 [1,4,3,2,5] => 1 [1,4,3,5,2] => 1 [1,4,5,2,3] => 1 [1,4,5,3,2] => 2 [1,5,2,3,4] => 0 [1,5,2,4,3] => 1 [1,5,3,2,4] => 1 [1,5,3,4,2] => 2 [1,5,4,2,3] => 2 [1,5,4,3,2] => 2 [2,1,3,4,5] => 0 [2,1,3,5,4] => 0 [2,1,4,3,5] => 0 [2,1,4,5,3] => 0 [2,1,5,3,4] => 0 [2,1,5,4,3] => 1 [2,3,1,4,5] => 0 [2,3,1,5,4] => 0 [2,3,4,1,5] => 0 [2,3,4,5,1] => 0 [2,3,5,1,4] => 0 [2,3,5,4,1] => 1 [2,4,1,3,5] => 0 [2,4,1,5,3] => 0 [2,4,3,1,5] => 1 [2,4,3,5,1] => 1 [2,4,5,1,3] => 1 [2,4,5,3,1] => 2 [2,5,1,3,4] => 0 [2,5,1,4,3] => 1 [2,5,3,1,4] => 1 [2,5,3,4,1] => 2 [2,5,4,1,3] => 2 [2,5,4,3,1] => 2 [3,1,2,4,5] => 0 [3,1,2,5,4] => 0 [3,1,4,2,5] => 0 [3,1,4,5,2] => 0 [3,1,5,2,4] => 0 [3,1,5,4,2] => 1 [3,2,1,4,5] => 1 [3,2,1,5,4] => 1 [3,2,4,1,5] => 1 [3,2,4,5,1] => 1 [3,2,5,1,4] => 1 [3,2,5,4,1] => 2 [3,4,1,2,5] => 1 [3,4,1,5,2] => 1 [3,4,2,1,5] => 2 [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] => 3 [3,5,4,1,2] => 2 [3,5,4,2,1] => 3 [4,1,2,3,5] => 0 [4,1,2,5,3] => 0 [4,1,3,2,5] => 1 [4,1,3,5,2] => 1 [4,1,5,2,3] => 1 [4,1,5,3,2] => 2 [4,2,1,3,5] => 1 [4,2,1,5,3] => 1 [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] => 2 [4,3,1,5,2] => 2 [4,3,2,1,5] => 2 [4,3,2,5,1] => 2 [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] => 3 [4,5,3,1,2] => 3 [4,5,3,2,1] => 4 [5,1,2,3,4] => 0 [5,1,2,4,3] => 1 [5,1,3,2,4] => 1 [5,1,3,4,2] => 2 [5,1,4,2,3] => 2 [5,1,4,3,2] => 2 [5,2,1,3,4] => 1 [5,2,1,4,3] => 2 [5,2,3,1,4] => 2 [5,2,3,4,1] => 3 [5,2,4,1,3] => 3 [5,2,4,3,1] => 3 [5,3,1,2,4] => 2 [5,3,1,4,2] => 3 [5,3,2,1,4] => 2 [5,3,2,4,1] => 3 [5,3,4,1,2] => 3 [5,3,4,2,1] => 4 [5,4,1,2,3] => 2 [5,4,1,3,2] => 3 [5,4,2,1,3] => 3 [5,4,2,3,1] => 4 [5,4,3,1,2] => 4 [5,4,3,2,1] => 5 [1,2,3,4,5,6] => 0 [1,2,3,4,6,5] => 0 [1,2,3,5,4,6] => 0 [1,2,3,5,6,4] => 0 [1,2,3,6,4,5] => 0 [1,2,3,6,5,4] => 1 [1,2,4,3,5,6] => 0 [1,2,4,3,6,5] => 0 [1,2,4,5,3,6] => 0 [1,2,4,5,6,3] => 0 [1,2,4,6,3,5] => 0 [1,2,4,6,5,3] => 1 [1,2,5,3,4,6] => 0 [1,2,5,3,6,4] => 0 [1,2,5,4,3,6] => 1 [1,2,5,4,6,3] => 1 [1,2,5,6,3,4] => 1 [1,2,5,6,4,3] => 2 [1,2,6,3,4,5] => 0 [1,2,6,3,5,4] => 1 [1,2,6,4,3,5] => 1 [1,2,6,4,5,3] => 2 [1,2,6,5,3,4] => 2 [1,2,6,5,4,3] => 2 [1,3,2,4,5,6] => 0 [1,3,2,4,6,5] => 0 [1,3,2,5,4,6] => 0 [1,3,2,5,6,4] => 0 [1,3,2,6,4,5] => 0 [1,3,2,6,5,4] => 1 [1,3,4,2,5,6] => 0 [1,3,4,2,6,5] => 0 [1,3,4,5,2,6] => 0 [1,3,4,5,6,2] => 0 [1,3,4,6,2,5] => 0 [1,3,4,6,5,2] => 1 [1,3,5,2,4,6] => 0 [1,3,5,2,6,4] => 0 [1,3,5,4,2,6] => 1 [1,3,5,4,6,2] => 1 [1,3,5,6,2,4] => 1 [1,3,5,6,4,2] => 2 [1,3,6,2,4,5] => 0 [1,3,6,2,5,4] => 1 [1,3,6,4,2,5] => 1 [1,3,6,4,5,2] => 2 [1,3,6,5,2,4] => 2 [1,3,6,5,4,2] => 2 [1,4,2,3,5,6] => 0 [1,4,2,3,6,5] => 0 [1,4,2,5,3,6] => 0 [1,4,2,5,6,3] => 0 [1,4,2,6,3,5] => 0 [1,4,2,6,5,3] => 1 [1,4,3,2,5,6] => 1 [1,4,3,2,6,5] => 1 [1,4,3,5,2,6] => 1 [1,4,3,5,6,2] => 1 [1,4,3,6,2,5] => 1 [1,4,3,6,5,2] => 2 [1,4,5,2,3,6] => 1 [1,4,5,2,6,3] => 1 [1,4,5,3,2,6] => 2 [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] => 3 [1,4,6,5,2,3] => 2 [1,4,6,5,3,2] => 3 [1,5,2,3,4,6] => 0 [1,5,2,3,6,4] => 0 [1,5,2,4,3,6] => 1 [1,5,2,4,6,3] => 1 [1,5,2,6,3,4] => 1 [1,5,2,6,4,3] => 2 [1,5,3,2,4,6] => 1 [1,5,3,2,6,4] => 1 [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] => 2 [1,5,4,2,6,3] => 2 [1,5,4,3,2,6] => 2 [1,5,4,3,6,2] => 2 [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] => 3 [1,5,6,4,2,3] => 3 [1,5,6,4,3,2] => 4 [1,6,2,3,4,5] => 0 [1,6,2,3,5,4] => 1 [1,6,2,4,3,5] => 1 [1,6,2,4,5,3] => 2 [1,6,2,5,3,4] => 2 [1,6,2,5,4,3] => 2 [1,6,3,2,4,5] => 1 [1,6,3,2,5,4] => 2 [1,6,3,4,2,5] => 2 [1,6,3,4,5,2] => 3 [1,6,3,5,2,4] => 3 [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] => 2 [1,6,4,3,5,2] => 3 [1,6,4,5,2,3] => 3 [1,6,4,5,3,2] => 4 [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] => 4 [1,6,5,4,2,3] => 4 [1,6,5,4,3,2] => 5 [2,1,3,4,5,6] => 0 [2,1,3,4,6,5] => 0 [2,1,3,5,4,6] => 0 [2,1,3,5,6,4] => 0 [2,1,3,6,4,5] => 0 [2,1,3,6,5,4] => 1 [2,1,4,3,5,6] => 0 [2,1,4,3,6,5] => 0 [2,1,4,5,3,6] => 0 [2,1,4,5,6,3] => 0 [2,1,4,6,3,5] => 0 [2,1,4,6,5,3] => 1 [2,1,5,3,4,6] => 0 [2,1,5,3,6,4] => 0 [2,1,5,4,3,6] => 1 [2,1,5,4,6,3] => 1 [2,1,5,6,3,4] => 1 [2,1,5,6,4,3] => 2 [2,1,6,3,4,5] => 0 [2,1,6,3,5,4] => 1 [2,1,6,4,3,5] => 1 [2,1,6,4,5,3] => 2 [2,1,6,5,3,4] => 2 [2,1,6,5,4,3] => 2 [2,3,1,4,5,6] => 0 [2,3,1,4,6,5] => 0 [2,3,1,5,4,6] => 0 [2,3,1,5,6,4] => 0 [2,3,1,6,4,5] => 0 [2,3,1,6,5,4] => 1 [2,3,4,1,5,6] => 0 [2,3,4,1,6,5] => 0 [2,3,4,5,1,6] => 0 [2,3,4,5,6,1] => 0 [2,3,4,6,1,5] => 0 [2,3,4,6,5,1] => 1 [2,3,5,1,4,6] => 0 [2,3,5,1,6,4] => 0 [2,3,5,4,1,6] => 1 [2,3,5,4,6,1] => 1 [2,3,5,6,1,4] => 1 [2,3,5,6,4,1] => 2 [2,3,6,1,4,5] => 0 [2,3,6,1,5,4] => 1 [2,3,6,4,1,5] => 1 [2,3,6,4,5,1] => 2 [2,3,6,5,1,4] => 2 [2,3,6,5,4,1] => 2 [2,4,1,3,5,6] => 0 [2,4,1,3,6,5] => 0 [2,4,1,5,3,6] => 0 [2,4,1,5,6,3] => 0 [2,4,1,6,3,5] => 0 [2,4,1,6,5,3] => 1 [2,4,3,1,5,6] => 1 [2,4,3,1,6,5] => 1 [2,4,3,5,1,6] => 1 [2,4,3,5,6,1] => 1 [2,4,3,6,1,5] => 1 [2,4,3,6,5,1] => 2 [2,4,5,1,3,6] => 1 [2,4,5,1,6,3] => 1 [2,4,5,3,1,6] => 2 [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] => 3 [2,4,6,5,1,3] => 2 [2,4,6,5,3,1] => 3 [2,5,1,3,4,6] => 0 [2,5,1,3,6,4] => 0 [2,5,1,4,3,6] => 1 [2,5,1,6,3,4] => 1 [2,5,1,6,4,3] => 2 [2,5,3,1,4,6] => 1 [2,5,3,1,6,4] => 1 [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] => 2 [2,5,4,1,6,3] => 2 [2,5,4,3,1,6] => 2 [2,5,4,3,6,1] => 2 [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] => 3 [2,5,6,4,1,3] => 3 [2,5,6,4,3,1] => 4 [2,6,1,3,4,5] => 0 [2,6,1,3,5,4] => 1 [2,6,1,4,3,5] => 1 [2,6,1,4,5,3] => 2 [2,6,1,5,3,4] => 2 [2,6,1,5,4,3] => 2 [2,6,3,1,4,5] => 1 [2,6,3,4,1,5] => 2 [2,6,3,4,5,1] => 3 [2,6,3,5,1,4] => 3 [2,6,3,5,4,1] => 3 [2,6,4,1,3,5] => 2 [2,6,4,1,5,3] => 3 [2,6,4,3,5,1] => 3 [2,6,4,5,1,3] => 3 [2,6,4,5,3,1] => 4 [2,6,5,1,4,3] => 3 [2,6,5,3,1,4] => 3 [2,6,5,3,4,1] => 4 [2,6,5,4,1,3] => 4 [2,6,5,4,3,1] => 5 [3,1,2,4,5,6] => 0 [3,1,2,4,6,5] => 0 [3,1,2,5,4,6] => 0 [3,1,2,5,6,4] => 0 [3,1,2,6,4,5] => 0 [3,1,2,6,5,4] => 1 [3,1,4,2,5,6] => 0 [3,1,4,2,6,5] => 0 [3,1,4,5,2,6] => 0 [3,1,4,5,6,2] => 0 [3,1,4,6,2,5] => 0 [3,1,4,6,5,2] => 1 [3,1,5,2,4,6] => 0 [3,1,5,2,6,4] => 0 [3,1,5,4,2,6] => 1 [3,1,5,4,6,2] => 1 [3,1,5,6,2,4] => 1 [3,1,5,6,4,2] => 2 [3,1,6,2,4,5] => 0 [3,1,6,2,5,4] => 1 [3,1,6,4,2,5] => 1 [3,1,6,4,5,2] => 2 [3,1,6,5,2,4] => 2 [3,1,6,5,4,2] => 2 [3,2,1,4,5,6] => 1 [3,2,1,4,6,5] => 1 [3,2,1,5,4,6] => 1 [3,2,1,5,6,4] => 1 [3,2,1,6,4,5] => 1 [3,2,1,6,5,4] => 2 [3,2,4,1,5,6] => 1 [3,2,4,1,6,5] => 1 [3,2,4,5,1,6] => 1 [3,2,4,5,6,1] => 1 [3,2,4,6,1,5] => 1 [3,2,4,6,5,1] => 2 [3,2,5,1,4,6] => 1 [3,2,5,1,6,4] => 1 [3,2,5,4,1,6] => 2 [3,2,5,4,6,1] => 2 [3,2,5,6,1,4] => 2 [3,2,5,6,4,1] => 3 [3,2,6,1,4,5] => 1 [3,2,6,1,5,4] => 2 [3,2,6,4,1,5] => 2 [3,2,6,4,5,1] => 3 [3,2,6,5,1,4] => 3 [3,2,6,5,4,1] => 3 [3,4,1,2,5,6] => 1 [3,4,1,2,6,5] => 1 [3,4,1,5,2,6] => 1 [3,4,1,5,6,2] => 1 [3,4,1,6,2,5] => 1 [3,4,1,6,5,2] => 2 [3,4,2,1,5,6] => 2 [3,4,2,1,6,5] => 2 [3,4,2,5,1,6] => 2 [3,4,2,5,6,1] => 2 [3,4,2,6,1,5] => 2 [3,4,2,6,5,1] => 3 [3,4,5,1,2,6] => 1 [3,4,5,1,6,2] => 1 [3,4,5,2,1,6] => 2 [3,4,5,2,6,1] => 2 [3,4,5,6,1,2] => 2 [3,4,5,6,2,1] => 3 [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] => 3 [3,4,6,5,1,2] => 3 [3,4,6,5,2,1] => 4 [3,5,1,2,4,6] => 1 [3,5,1,2,6,4] => 1 [3,5,1,4,2,6] => 2 [3,5,1,4,6,2] => 2 [3,5,1,6,2,4] => 2 [3,5,1,6,4,2] => 3 [3,5,2,1,4,6] => 2 [3,5,2,1,6,4] => 2 [3,5,2,4,1,6] => 3 [3,5,2,4,6,1] => 3 [3,5,2,6,1,4] => 3 [3,5,2,6,4,1] => 4 [3,5,4,1,2,6] => 2 [3,5,4,1,6,2] => 2 [3,5,4,2,1,6] => 3 [3,5,4,2,6,1] => 3 [3,5,4,6,1,2] => 3 [3,5,4,6,2,1] => 4 [3,5,6,1,2,4] => 2 [3,5,6,1,4,2] => 3 [3,5,6,2,1,4] => 3 [3,5,6,2,4,1] => 4 [3,5,6,4,1,2] => 4 [3,5,6,4,2,1] => 5 [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] => 3 [3,6,1,5,2,4] => 3 [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] => 3 [3,6,2,4,5,1] => 4 [3,6,2,5,1,4] => 4 [3,6,2,5,4,1] => 4 [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] => 4 [3,6,4,5,1,2] => 4 [3,6,4,5,2,1] => 5 [3,6,5,1,2,4] => 3 [3,6,5,1,4,2] => 4 [3,6,5,2,1,4] => 4 [3,6,5,2,4,1] => 5 [3,6,5,4,1,2] => 5 [3,6,5,4,2,1] => 6 [4,1,2,3,5,6] => 0 [4,1,2,3,6,5] => 0 [4,1,2,5,3,6] => 0 [4,1,2,5,6,3] => 0 [4,1,2,6,3,5] => 0 [4,1,2,6,5,3] => 1 [4,1,3,2,5,6] => 1 [4,1,3,2,6,5] => 1 [4,1,3,5,2,6] => 1 [4,1,3,5,6,2] => 1 [4,1,3,6,2,5] => 1 [4,1,3,6,5,2] => 2 [4,1,5,2,3,6] => 1 [4,1,5,2,6,3] => 1 [4,1,5,3,2,6] => 2 [4,1,5,3,6,2] => 2 [4,1,5,6,2,3] => 1 [4,1,5,6,3,2] => 2 [4,1,6,2,3,5] => 1 [4,1,6,2,5,3] => 2 [4,1,6,3,2,5] => 2 [4,1,6,3,5,2] => 3 [4,1,6,5,2,3] => 2 [4,1,6,5,3,2] => 3 [4,2,1,3,5,6] => 1 [4,2,1,3,6,5] => 1 [4,2,1,5,3,6] => 1 [4,2,1,5,6,3] => 1 [4,2,1,6,3,5] => 1 [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] => 2 [4,2,3,5,6,1] => 2 [4,2,3,6,1,5] => 2 [4,2,3,6,5,1] => 3 [4,2,5,1,3,6] => 2 [4,2,5,1,6,3] => 2 [4,2,5,3,1,6] => 3 [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] => 4 [4,2,6,5,1,3] => 3 [4,2,6,5,3,1] => 4 [4,3,1,2,5,6] => 2 [4,3,1,2,6,5] => 2 [4,3,1,5,2,6] => 2 [4,3,1,5,6,2] => 2 [4,3,1,6,2,5] => 2 [4,3,1,6,5,2] => 3 [4,3,2,1,5,6] => 2 [4,3,2,1,6,5] => 2 [4,3,2,5,1,6] => 2 [4,3,2,5,6,1] => 2 [4,3,2,6,1,5] => 2 [4,3,2,6,5,1] => 3 [4,3,5,1,2,6] => 2 [4,3,5,1,6,2] => 2 [4,3,5,2,1,6] => 3 [4,3,5,2,6,1] => 3 [4,3,5,6,1,2] => 3 [4,3,5,6,2,1] => 4 [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] => 4 [4,3,6,5,1,2] => 4 [4,3,6,5,2,1] => 4 [4,5,1,2,3,6] => 1 [4,5,1,2,6,3] => 1 [4,5,1,3,2,6] => 2 [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] => 2 [4,5,2,1,6,3] => 2 [4,5,2,3,1,6] => 3 [4,5,2,3,6,1] => 3 [4,5,2,6,1,3] => 3 [4,5,2,6,3,1] => 4 [4,5,3,1,2,6] => 3 [4,5,3,1,6,2] => 3 [4,5,3,2,1,6] => 4 [4,5,3,2,6,1] => 4 [4,5,3,6,1,2] => 4 [4,5,3,6,2,1] => 5 [4,5,6,1,2,3] => 3 [4,5,6,1,3,2] => 4 [4,5,6,2,1,3] => 4 [4,5,6,2,3,1] => 5 [4,5,6,3,1,2] => 4 [4,5,6,3,2,1] => 5 [4,6,1,2,3,5] => 1 [4,6,1,2,5,3] => 2 [4,6,1,3,5,2] => 3 [4,6,1,5,2,3] => 3 [4,6,1,5,3,2] => 4 [4,6,2,1,3,5] => 2 [4,6,2,1,5,3] => 3 [4,6,2,3,1,5] => 3 [4,6,2,3,5,1] => 4 [4,6,2,5,1,3] => 4 [4,6,2,5,3,1] => 4 [4,6,3,1,2,5] => 3 [4,6,3,1,5,2] => 4 [4,6,3,2,1,5] => 4 [4,6,3,2,5,1] => 5 [4,6,3,5,1,2] => 5 [4,6,3,5,2,1] => 5 [4,6,5,1,2,3] => 4 [4,6,5,1,3,2] => 4 [4,6,5,2,1,3] => 5 [4,6,5,2,3,1] => 5 [4,6,5,3,1,2] => 5 [4,6,5,3,2,1] => 6 [5,1,2,3,4,6] => 0 [5,1,2,3,6,4] => 0 [5,1,2,4,3,6] => 1 [5,1,2,4,6,3] => 1 [5,1,2,6,3,4] => 1 [5,1,2,6,4,3] => 2 [5,1,3,2,4,6] => 1 [5,1,3,2,6,4] => 1 [5,1,3,4,2,6] => 2 [5,1,3,4,6,2] => 2 [5,1,3,6,2,4] => 2 [5,1,3,6,4,2] => 3 [5,1,4,2,3,6] => 2 [5,1,4,2,6,3] => 2 [5,1,4,3,2,6] => 2 [5,1,4,6,3,2] => 3 [5,1,6,2,3,4] => 1 [5,1,6,2,4,3] => 2 [5,1,6,3,2,4] => 2 [5,1,6,3,4,2] => 3 [5,1,6,4,2,3] => 3 [5,1,6,4,3,2] => 4 [5,2,1,3,4,6] => 1 [5,2,1,3,6,4] => 1 [5,2,1,4,3,6] => 2 [5,2,1,4,6,3] => 2 [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] => 3 [5,2,3,4,6,1] => 3 [5,2,3,6,1,4] => 3 [5,2,3,6,4,1] => 4 [5,2,4,1,3,6] => 3 [5,2,4,1,6,3] => 3 [5,2,4,3,1,6] => 3 [5,2,4,3,6,1] => 3 [5,2,4,6,1,3] => 3 [5,2,4,6,3,1] => 4 [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] => 4 [5,2,6,4,1,3] => 4 [5,2,6,4,3,1] => 5 [5,3,1,2,4,6] => 2 [5,3,1,2,6,4] => 2 [5,3,1,4,2,6] => 3 [5,3,1,4,6,2] => 3 [5,3,1,6,2,4] => 3 [5,3,1,6,4,2] => 4 [5,3,2,1,4,6] => 2 [5,3,2,1,6,4] => 2 [5,3,2,4,1,6] => 3 [5,3,2,4,6,1] => 3 [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] => 4 [5,3,4,6,1,2] => 4 [5,3,4,6,2,1] => 5 [5,3,6,1,2,4] => 3 [5,3,6,1,4,2] => 4 [5,3,6,2,1,4] => 4 [5,3,6,2,4,1] => 4 [5,3,6,4,1,2] => 5 [5,3,6,4,2,1] => 5 [5,4,1,2,3,6] => 2 [5,4,1,2,6,3] => 2 [5,4,1,3,2,6] => 3 [5,4,1,6,2,3] => 3 [5,4,1,6,3,2] => 4 [5,4,2,1,3,6] => 3 [5,4,2,1,6,3] => 3 [5,4,2,3,1,6] => 4 [5,4,2,3,6,1] => 4 [5,4,2,6,1,3] => 4 [5,4,2,6,3,1] => 5 [5,4,3,1,2,6] => 4 [5,4,3,1,6,2] => 4 [5,4,3,2,1,6] => 5 [5,4,3,2,6,1] => 5 [5,4,3,6,1,2] => 5 [5,4,3,6,2,1] => 6 [5,4,6,1,2,3] => 4 [5,4,6,1,3,2] => 5 [5,4,6,2,1,3] => 4 [5,4,6,2,3,1] => 5 [5,4,6,3,1,2] => 5 [5,4,6,3,2,1] => 6 [5,6,1,2,3,4] => 2 [5,6,1,2,4,3] => 3 [5,6,1,3,2,4] => 3 [5,6,1,3,4,2] => 4 [5,6,1,4,2,3] => 4 [5,6,1,4,3,2] => 5 [5,6,2,1,3,4] => 3 [5,6,2,1,4,3] => 4 [5,6,2,3,1,4] => 4 [5,6,2,3,4,1] => 4 [5,6,2,4,1,3] => 5 [5,6,2,4,3,1] => 5 [5,6,3,1,2,4] => 4 [5,6,3,1,4,2] => 5 [5,6,3,2,1,4] => 5 [5,6,3,2,4,1] => 5 [5,6,3,4,1,2] => 6 [5,6,3,4,2,1] => 6 [5,6,4,1,2,3] => 4 [5,6,4,1,3,2] => 5 [5,6,4,2,1,3] => 5 [5,6,4,2,3,1] => 6 [5,6,4,3,1,2] => 6 [5,6,4,3,2,1] => 7 [6,1,2,3,4,5] => 0 [6,1,2,3,5,4] => 1 [6,1,2,4,3,5] => 1 [6,1,2,4,5,3] => 2 [6,1,2,5,3,4] => 2 [6,1,2,5,4,3] => 2 [6,1,3,2,4,5] => 1 [6,1,3,2,5,4] => 2 [6,1,3,4,2,5] => 2 [6,1,3,4,5,2] => 3 [6,1,3,5,2,4] => 3 [6,1,3,5,4,2] => 3 [6,1,4,2,3,5] => 2 [6,1,4,2,5,3] => 3 [6,1,4,3,2,5] => 2 [6,1,4,3,5,2] => 3 [6,1,4,5,2,3] => 3 [6,1,4,5,3,2] => 4 [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] => 4 [6,1,5,4,2,3] => 4 [6,1,5,4,3,2] => 5 [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] => 3 [6,2,1,5,3,4] => 3 [6,2,1,5,4,3] => 3 [6,2,3,1,4,5] => 2 [6,2,3,1,5,4] => 3 [6,2,3,4,1,5] => 3 [6,2,3,4,5,1] => 4 [6,2,3,5,1,4] => 4 [6,2,3,5,4,1] => 4 [6,2,4,1,3,5] => 3 [6,2,4,1,5,3] => 4 [6,2,4,3,1,5] => 3 [6,2,4,3,5,1] => 4 [6,2,4,5,1,3] => 4 [6,2,4,5,3,1] => 5 [6,2,5,1,3,4] => 3 [6,2,5,1,4,3] => 4 [6,2,5,3,1,4] => 4 [6,2,5,3,4,1] => 5 [6,2,5,4,1,3] => 5 [6,2,5,4,3,1] => 6 [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] => 4 [6,3,1,5,2,4] => 4 [6,3,1,5,4,2] => 4 [6,3,2,1,4,5] => 2 [6,3,2,1,5,4] => 3 [6,3,2,4,1,5] => 3 [6,3,2,4,5,1] => 4 [6,3,2,5,1,4] => 4 [6,3,2,5,4,1] => 4 [6,3,4,1,2,5] => 3 [6,3,4,1,5,2] => 4 [6,3,4,2,1,5] => 4 [6,3,4,2,5,1] => 5 [6,3,4,5,1,2] => 4 [6,3,4,5,2,1] => 5 [6,3,5,1,2,4] => 4 [6,3,5,1,4,2] => 4 [6,3,5,2,1,4] => 5 [6,3,5,2,4,1] => 5 [6,3,5,4,1,2] => 5 [6,3,5,4,2,1] => 6 [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] => 4 [6,4,1,5,2,3] => 4 [6,4,1,5,3,2] => 5 [6,4,2,1,3,5] => 3 [6,4,2,1,5,3] => 4 [6,4,2,3,1,5] => 4 [6,4,2,3,5,1] => 5 [6,4,2,5,1,3] => 4 [6,4,2,5,3,1] => 5 [6,4,3,1,2,5] => 4 [6,4,3,1,5,2] => 5 [6,4,3,2,1,5] => 5 [6,4,3,2,5,1] => 6 [6,4,3,5,1,2] => 5 [6,4,3,5,2,1] => 6 [6,4,5,1,2,3] => 5 [6,4,5,1,3,2] => 5 [6,4,5,2,1,3] => 5 [6,4,5,2,3,1] => 6 [6,4,5,3,1,2] => 6 [6,4,5,3,2,1] => 7 [6,5,1,2,3,4] => 3 [6,5,1,2,4,3] => 4 [6,5,1,3,2,4] => 4 [6,5,1,3,4,2] => 5 [6,5,1,4,2,3] => 5 [6,5,1,4,3,2] => 6 [6,5,2,1,3,4] => 4 [6,5,2,1,4,3] => 4 [6,5,2,3,1,4] => 5 [6,5,2,3,4,1] => 5 [6,5,2,4,1,3] => 5 [6,5,2,4,3,1] => 6 [6,5,3,1,2,4] => 5 [6,5,3,1,4,2] => 5 [6,5,3,2,1,4] => 6 ----------------------------------------------------------------------------- Created: Feb 13, 2013 at 23:53 by Sara Billey ----------------------------------------------------------------------------- Last Updated: Mar 31, 2019 at 21:08 by Martin Rubey