***************************************************************************** * 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: St000055 ----------------------------------------------------------------------------- Collection: Permutations ----------------------------------------------------------------------------- Description: The inversion sum of a permutation. A pair $a < b$ is an inversion of a permutation $\pi$ if $\pi(a) > \pi(b)$. The inversion sum is given by $\sum(b-a)$ over all inversions of $\pi$. This is also half of the metric associated with Spearmans coefficient of association $\rho$, $\sum_i (\pi_i - i)^2$, see [5]. This is also equal to the total number of occurrences of the classical permutation patterns $[2,1], [2, 3, 1], [3, 1, 2]$, and $[3, 2, 1]$, see [2]. This is also equal to the rank of the permutation inside the alternating sign matrix lattice, see references [2] and [3]. This lattice is the MacNeille completion of the strong Bruhat order on the symmetric group [1], which means it is the smallest lattice containing the Bruhat order as a subposet. This is a distributive lattice, so the rank of each element is given by the cardinality of the associated order ideal. The rank is calculated by summing the entries of the corresponding ''monotone triangle'' and subtracting $\binom{n+2}{3}$, which is the sum of the entries of the monotone triangle corresponding to the identity permutation of $n$. This is also the number of bigrassmannian permutations (that is, permutations with exactly one left descent and one right descent) below a given permutation $\pi$ in Bruhat order, see Theorem 1 of [6]. ----------------------------------------------------------------------------- References: [1] Lascoux, A., Schützenberger, M.-P. Treillis et bases des groupes de Coxeter [[MathSciNet:1395667]] [2] Sack, J., Úlfarsson, H. Refined inversion statistics on permutations [[MathSciNet:2880660]] [[arXiv:1106.1995]] [3] Striker, J. A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux [[MathSciNet:2794039]] [4] a(n) = the total number of permutations (m(1),m(2),m(3)...m(j)) of (1,2,3,...,j) where n = 1*m(1) + 2*m(2) + 3*m(3) + ...+j*m(j), where j is over all positive integers. [[OEIS:A135298]] [5] Diaconis, P., Graham, R. L. Spearman's footrule as a measure of disarray [[MathSciNet:0652736]] [6] Kobayashi, M. Enumeration of bigrassmannian permutations below a permutation in Bruhat order [[arXiv:1005.3335]] ----------------------------------------------------------------------------- Code: def statistic(pi): return sum( inv[1]-inv[0] for inv in pi.inversions() ) def statistic_alternative(perm): pmatrix = perm.to_matrix() w = AlternatingSignMatrix(pmatrix).to_monotone_triangle() counter = -binomial(len(perm)+2,3) for k in [0..(len(w)-1)]: for j in [0..(len(w[k])-1)]: counter=counter+w[k][j] return counter ----------------------------------------------------------------------------- Statistic values: [1] => 0 [1,2] => 0 [2,1] => 1 [1,2,3] => 0 [1,3,2] => 1 [2,1,3] => 1 [2,3,1] => 3 [3,1,2] => 3 [3,2,1] => 4 [1,2,3,4] => 0 [1,2,4,3] => 1 [1,3,2,4] => 1 [1,3,4,2] => 3 [1,4,2,3] => 3 [1,4,3,2] => 4 [2,1,3,4] => 1 [2,1,4,3] => 2 [2,3,1,4] => 3 [2,3,4,1] => 6 [2,4,1,3] => 5 [2,4,3,1] => 7 [3,1,2,4] => 3 [3,1,4,2] => 5 [3,2,1,4] => 4 [3,2,4,1] => 7 [3,4,1,2] => 8 [3,4,2,1] => 9 [4,1,2,3] => 6 [4,1,3,2] => 7 [4,2,1,3] => 7 [4,2,3,1] => 9 [4,3,1,2] => 9 [4,3,2,1] => 10 [1,2,3,4,5] => 0 [1,2,3,5,4] => 1 [1,2,4,3,5] => 1 [1,2,4,5,3] => 3 [1,2,5,3,4] => 3 [1,2,5,4,3] => 4 [1,3,2,4,5] => 1 [1,3,2,5,4] => 2 [1,3,4,2,5] => 3 [1,3,4,5,2] => 6 [1,3,5,2,4] => 5 [1,3,5,4,2] => 7 [1,4,2,3,5] => 3 [1,4,2,5,3] => 5 [1,4,3,2,5] => 4 [1,4,3,5,2] => 7 [1,4,5,2,3] => 8 [1,4,5,3,2] => 9 [1,5,2,3,4] => 6 [1,5,2,4,3] => 7 [1,5,3,2,4] => 7 [1,5,3,4,2] => 9 [1,5,4,2,3] => 9 [1,5,4,3,2] => 10 [2,1,3,4,5] => 1 [2,1,3,5,4] => 2 [2,1,4,3,5] => 2 [2,1,4,5,3] => 4 [2,1,5,3,4] => 4 [2,1,5,4,3] => 5 [2,3,1,4,5] => 3 [2,3,1,5,4] => 4 [2,3,4,1,5] => 6 [2,3,4,5,1] => 10 [2,3,5,1,4] => 8 [2,3,5,4,1] => 11 [2,4,1,3,5] => 5 [2,4,1,5,3] => 7 [2,4,3,1,5] => 7 [2,4,3,5,1] => 11 [2,4,5,1,3] => 11 [2,4,5,3,1] => 13 [2,5,1,3,4] => 8 [2,5,1,4,3] => 9 [2,5,3,1,4] => 10 [2,5,3,4,1] => 13 [2,5,4,1,3] => 12 [2,5,4,3,1] => 14 [3,1,2,4,5] => 3 [3,1,2,5,4] => 4 [3,1,4,2,5] => 5 [3,1,4,5,2] => 8 [3,1,5,2,4] => 7 [3,1,5,4,2] => 9 [3,2,1,4,5] => 4 [3,2,1,5,4] => 5 [3,2,4,1,5] => 7 [3,2,4,5,1] => 11 [3,2,5,1,4] => 9 [3,2,5,4,1] => 12 [3,4,1,2,5] => 8 [3,4,1,5,2] => 11 [3,4,2,1,5] => 9 [3,4,2,5,1] => 13 [3,4,5,1,2] => 15 [3,4,5,2,1] => 16 [3,5,1,2,4] => 11 [3,5,1,4,2] => 13 [3,5,2,1,4] => 12 [3,5,2,4,1] => 15 [3,5,4,1,2] => 16 [3,5,4,2,1] => 17 [4,1,2,3,5] => 6 [4,1,2,5,3] => 8 [4,1,3,2,5] => 7 [4,1,3,5,2] => 10 [4,1,5,2,3] => 11 [4,1,5,3,2] => 12 [4,2,1,3,5] => 7 [4,2,1,5,3] => 9 [4,2,3,1,5] => 9 [4,2,3,5,1] => 13 [4,2,5,1,3] => 13 [4,2,5,3,1] => 15 [4,3,1,2,5] => 9 [4,3,1,5,2] => 12 [4,3,2,1,5] => 10 [4,3,2,5,1] => 14 [4,3,5,1,2] => 16 [4,3,5,2,1] => 17 [4,5,1,2,3] => 15 [4,5,1,3,2] => 16 [4,5,2,1,3] => 16 [4,5,2,3,1] => 18 [4,5,3,1,2] => 18 [4,5,3,2,1] => 19 [5,1,2,3,4] => 10 [5,1,2,4,3] => 11 [5,1,3,2,4] => 11 [5,1,3,4,2] => 13 [5,1,4,2,3] => 13 [5,1,4,3,2] => 14 [5,2,1,3,4] => 11 [5,2,1,4,3] => 12 [5,2,3,1,4] => 13 [5,2,3,4,1] => 16 [5,2,4,1,3] => 15 [5,2,4,3,1] => 17 [5,3,1,2,4] => 13 [5,3,1,4,2] => 15 [5,3,2,1,4] => 14 [5,3,2,4,1] => 17 [5,3,4,1,2] => 18 [5,3,4,2,1] => 19 [5,4,1,2,3] => 16 [5,4,1,3,2] => 17 [5,4,2,1,3] => 17 [5,4,2,3,1] => 19 [5,4,3,1,2] => 19 [5,4,3,2,1] => 20 [1,2,3,4,5,6] => 0 [1,2,3,4,6,5] => 1 [1,2,3,5,4,6] => 1 [1,2,3,5,6,4] => 3 [1,2,3,6,4,5] => 3 [1,2,3,6,5,4] => 4 [1,2,4,3,5,6] => 1 [1,2,4,3,6,5] => 2 [1,2,4,5,3,6] => 3 [1,2,4,5,6,3] => 6 [1,2,4,6,3,5] => 5 [1,2,4,6,5,3] => 7 [1,2,5,3,4,6] => 3 [1,2,5,3,6,4] => 5 [1,2,5,4,3,6] => 4 [1,2,5,4,6,3] => 7 [1,2,5,6,3,4] => 8 [1,2,5,6,4,3] => 9 [1,2,6,3,4,5] => 6 [1,2,6,3,5,4] => 7 [1,2,6,4,3,5] => 7 [1,2,6,4,5,3] => 9 [1,2,6,5,3,4] => 9 [1,2,6,5,4,3] => 10 [1,3,2,4,5,6] => 1 [1,3,2,4,6,5] => 2 [1,3,2,5,4,6] => 2 [1,3,2,5,6,4] => 4 [1,3,2,6,4,5] => 4 [1,3,2,6,5,4] => 5 [1,3,4,2,5,6] => 3 [1,3,4,2,6,5] => 4 [1,3,4,5,2,6] => 6 [1,3,4,5,6,2] => 10 [1,3,4,6,2,5] => 8 [1,3,4,6,5,2] => 11 [1,3,5,2,4,6] => 5 [1,3,5,2,6,4] => 7 [1,3,5,4,2,6] => 7 [1,3,5,4,6,2] => 11 [1,3,5,6,2,4] => 11 [1,3,5,6,4,2] => 13 [1,3,6,2,4,5] => 8 [1,3,6,2,5,4] => 9 [1,3,6,4,2,5] => 10 [1,3,6,4,5,2] => 13 [1,3,6,5,2,4] => 12 [1,3,6,5,4,2] => 14 [1,4,2,3,5,6] => 3 [1,4,2,3,6,5] => 4 [1,4,2,5,3,6] => 5 [1,4,2,5,6,3] => 8 [1,4,2,6,3,5] => 7 [1,4,2,6,5,3] => 9 [1,4,3,2,5,6] => 4 [1,4,3,2,6,5] => 5 [1,4,3,5,2,6] => 7 [1,4,3,5,6,2] => 11 [1,4,3,6,2,5] => 9 [1,4,3,6,5,2] => 12 [1,4,5,2,3,6] => 8 [1,4,5,2,6,3] => 11 [1,4,5,3,2,6] => 9 [1,4,5,3,6,2] => 13 [1,4,5,6,2,3] => 15 [1,4,5,6,3,2] => 16 [1,4,6,2,3,5] => 11 [1,4,6,2,5,3] => 13 [1,4,6,3,2,5] => 12 [1,4,6,3,5,2] => 15 [1,4,6,5,2,3] => 16 [1,4,6,5,3,2] => 17 [1,5,2,3,4,6] => 6 [1,5,2,3,6,4] => 8 [1,5,2,4,3,6] => 7 [1,5,2,4,6,3] => 10 [1,5,2,6,3,4] => 11 [1,5,2,6,4,3] => 12 [1,5,3,2,4,6] => 7 [1,5,3,2,6,4] => 9 [1,5,3,4,2,6] => 9 [1,5,3,4,6,2] => 13 [1,5,3,6,2,4] => 13 [1,5,3,6,4,2] => 15 [1,5,4,2,3,6] => 9 [1,5,4,2,6,3] => 12 [1,5,4,3,2,6] => 10 [1,5,4,3,6,2] => 14 [1,5,4,6,2,3] => 16 [1,5,4,6,3,2] => 17 [1,5,6,2,3,4] => 15 [1,5,6,2,4,3] => 16 [1,5,6,3,2,4] => 16 [1,5,6,3,4,2] => 18 [1,5,6,4,2,3] => 18 [1,5,6,4,3,2] => 19 [1,6,2,3,4,5] => 10 [1,6,2,3,5,4] => 11 [1,6,2,4,3,5] => 11 [1,6,2,4,5,3] => 13 [1,6,2,5,3,4] => 13 [1,6,2,5,4,3] => 14 [1,6,3,2,4,5] => 11 [1,6,3,2,5,4] => 12 [1,6,3,4,2,5] => 13 [1,6,3,4,5,2] => 16 [1,6,3,5,2,4] => 15 [1,6,3,5,4,2] => 17 [1,6,4,2,3,5] => 13 [1,6,4,2,5,3] => 15 [1,6,4,3,2,5] => 14 [1,6,4,3,5,2] => 17 [1,6,4,5,2,3] => 18 [1,6,4,5,3,2] => 19 [1,6,5,2,3,4] => 16 [1,6,5,2,4,3] => 17 [1,6,5,3,2,4] => 17 [1,6,5,3,4,2] => 19 [1,6,5,4,2,3] => 19 [1,6,5,4,3,2] => 20 [2,1,3,4,5,6] => 1 [2,1,3,4,6,5] => 2 [2,1,3,5,4,6] => 2 [2,1,3,5,6,4] => 4 [2,1,3,6,4,5] => 4 [2,1,3,6,5,4] => 5 [2,1,4,3,5,6] => 2 [2,1,4,3,6,5] => 3 [2,1,4,5,3,6] => 4 [2,1,4,5,6,3] => 7 [2,1,4,6,3,5] => 6 [2,1,4,6,5,3] => 8 [2,1,5,3,4,6] => 4 [2,1,5,3,6,4] => 6 [2,1,5,4,3,6] => 5 [2,1,5,4,6,3] => 8 [2,1,5,6,3,4] => 9 [2,1,5,6,4,3] => 10 [2,1,6,3,4,5] => 7 [2,1,6,3,5,4] => 8 [2,1,6,4,3,5] => 8 [2,1,6,4,5,3] => 10 [2,1,6,5,3,4] => 10 [2,1,6,5,4,3] => 11 [2,3,1,4,5,6] => 3 [2,3,1,4,6,5] => 4 [2,3,1,5,4,6] => 4 [2,3,1,5,6,4] => 6 [2,3,1,6,4,5] => 6 [2,3,1,6,5,4] => 7 [2,3,4,1,5,6] => 6 [2,3,4,1,6,5] => 7 [2,3,4,5,1,6] => 10 [2,3,4,5,6,1] => 15 [2,3,4,6,1,5] => 12 [2,3,4,6,5,1] => 16 [2,3,5,1,4,6] => 8 [2,3,5,1,6,4] => 10 [2,3,5,4,1,6] => 11 [2,3,5,4,6,1] => 16 [2,3,5,6,1,4] => 15 [2,3,5,6,4,1] => 18 [2,3,6,1,4,5] => 11 [2,3,6,1,5,4] => 12 [2,3,6,4,1,5] => 14 [2,3,6,4,5,1] => 18 [2,3,6,5,1,4] => 16 [2,3,6,5,4,1] => 19 [2,4,1,3,5,6] => 5 [2,4,1,3,6,5] => 6 [2,4,1,5,3,6] => 7 [2,4,1,5,6,3] => 10 [2,4,1,6,3,5] => 9 [2,4,1,6,5,3] => 11 [2,4,3,1,5,6] => 7 [2,4,3,1,6,5] => 8 [2,4,3,5,1,6] => 11 [2,4,3,5,6,1] => 16 [2,4,3,6,1,5] => 13 [2,4,3,6,5,1] => 17 [2,4,5,1,3,6] => 11 [2,4,5,1,6,3] => 14 [2,4,5,3,1,6] => 13 [2,4,5,3,6,1] => 18 [2,4,5,6,1,3] => 19 [2,4,5,6,3,1] => 21 [2,4,6,1,3,5] => 14 [2,4,6,1,5,3] => 16 [2,4,6,3,1,5] => 16 [2,4,6,3,5,1] => 20 [2,4,6,5,1,3] => 20 [2,4,6,5,3,1] => 22 [2,5,1,3,4,6] => 8 [2,5,1,3,6,4] => 10 [2,5,1,4,3,6] => 9 [2,5,1,4,6,3] => 12 [2,5,1,6,3,4] => 13 [2,5,1,6,4,3] => 14 [2,5,3,1,4,6] => 10 [2,5,3,1,6,4] => 12 [2,5,3,4,1,6] => 13 [2,5,3,4,6,1] => 18 [2,5,3,6,1,4] => 17 [2,5,3,6,4,1] => 20 [2,5,4,1,3,6] => 12 [2,5,4,1,6,3] => 15 [2,5,4,3,1,6] => 14 [2,5,4,3,6,1] => 19 [2,5,4,6,1,3] => 20 [2,5,4,6,3,1] => 22 [2,5,6,1,3,4] => 18 [2,5,6,1,4,3] => 19 [2,5,6,3,1,4] => 20 [2,5,6,3,4,1] => 23 [2,5,6,4,1,3] => 22 [2,5,6,4,3,1] => 24 [2,6,1,3,4,5] => 12 [2,6,1,3,5,4] => 13 [2,6,1,4,3,5] => 13 [2,6,1,4,5,3] => 15 [2,6,1,5,3,4] => 15 [2,6,1,5,4,3] => 16 [2,6,3,1,4,5] => 14 [2,6,3,1,5,4] => 15 [2,6,3,4,1,5] => 17 [2,6,3,4,5,1] => 21 [2,6,3,5,1,4] => 19 [2,6,3,5,4,1] => 22 [2,6,4,1,3,5] => 16 [2,6,4,1,5,3] => 18 [2,6,4,3,1,5] => 18 [2,6,4,3,5,1] => 22 [2,6,4,5,1,3] => 22 [2,6,4,5,3,1] => 24 [2,6,5,1,3,4] => 19 [2,6,5,1,4,3] => 20 [2,6,5,3,1,4] => 21 [2,6,5,3,4,1] => 24 [2,6,5,4,1,3] => 23 [2,6,5,4,3,1] => 25 [3,1,2,4,5,6] => 3 [3,1,2,4,6,5] => 4 [3,1,2,5,4,6] => 4 [3,1,2,5,6,4] => 6 [3,1,2,6,4,5] => 6 [3,1,2,6,5,4] => 7 [3,1,4,2,5,6] => 5 [3,1,4,2,6,5] => 6 [3,1,4,5,2,6] => 8 [3,1,4,5,6,2] => 12 [3,1,4,6,2,5] => 10 [3,1,4,6,5,2] => 13 [3,1,5,2,4,6] => 7 [3,1,5,2,6,4] => 9 [3,1,5,4,2,6] => 9 [3,1,5,4,6,2] => 13 [3,1,5,6,2,4] => 13 [3,1,5,6,4,2] => 15 [3,1,6,2,4,5] => 10 [3,1,6,2,5,4] => 11 [3,1,6,4,2,5] => 12 [3,1,6,4,5,2] => 15 [3,1,6,5,2,4] => 14 [3,1,6,5,4,2] => 16 [3,2,1,4,5,6] => 4 [3,2,1,4,6,5] => 5 [3,2,1,5,4,6] => 5 [3,2,1,5,6,4] => 7 [3,2,1,6,4,5] => 7 [3,2,1,6,5,4] => 8 [3,2,4,1,5,6] => 7 [3,2,4,1,6,5] => 8 [3,2,4,5,1,6] => 11 [3,2,4,5,6,1] => 16 [3,2,4,6,1,5] => 13 [3,2,4,6,5,1] => 17 [3,2,5,1,4,6] => 9 [3,2,5,1,6,4] => 11 [3,2,5,4,1,6] => 12 [3,2,5,4,6,1] => 17 [3,2,5,6,1,4] => 16 [3,2,5,6,4,1] => 19 [3,2,6,1,4,5] => 12 [3,2,6,1,5,4] => 13 [3,2,6,4,1,5] => 15 [3,2,6,4,5,1] => 19 [3,2,6,5,1,4] => 17 [3,2,6,5,4,1] => 20 [3,4,1,2,5,6] => 8 [3,4,1,2,6,5] => 9 [3,4,1,5,2,6] => 11 [3,4,1,5,6,2] => 15 [3,4,1,6,2,5] => 13 [3,4,1,6,5,2] => 16 [3,4,2,1,5,6] => 9 [3,4,2,1,6,5] => 10 [3,4,2,5,1,6] => 13 [3,4,2,5,6,1] => 18 [3,4,2,6,1,5] => 15 [3,4,2,6,5,1] => 19 [3,4,5,1,2,6] => 15 [3,4,5,1,6,2] => 19 [3,4,5,2,1,6] => 16 [3,4,5,2,6,1] => 21 [3,4,5,6,1,2] => 24 [3,4,5,6,2,1] => 25 [3,4,6,1,2,5] => 18 [3,4,6,1,5,2] => 21 [3,4,6,2,1,5] => 19 [3,4,6,2,5,1] => 23 [3,4,6,5,1,2] => 25 [3,4,6,5,2,1] => 26 [3,5,1,2,4,6] => 11 [3,5,1,2,6,4] => 13 [3,5,1,4,2,6] => 13 [3,5,1,4,6,2] => 17 [3,5,1,6,2,4] => 17 [3,5,1,6,4,2] => 19 [3,5,2,1,4,6] => 12 [3,5,2,1,6,4] => 14 [3,5,2,4,1,6] => 15 [3,5,2,4,6,1] => 20 [3,5,2,6,1,4] => 19 [3,5,2,6,4,1] => 22 [3,5,4,1,2,6] => 16 [3,5,4,1,6,2] => 20 [3,5,4,2,1,6] => 17 [3,5,4,2,6,1] => 22 [3,5,4,6,1,2] => 25 [3,5,4,6,2,1] => 26 [3,5,6,1,2,4] => 22 [3,5,6,1,4,2] => 24 [3,5,6,2,1,4] => 23 [3,5,6,2,4,1] => 26 [3,5,6,4,1,2] => 27 [3,5,6,4,2,1] => 28 [3,6,1,2,4,5] => 15 [3,6,1,2,5,4] => 16 [3,6,1,4,2,5] => 17 [3,6,1,4,5,2] => 20 [3,6,1,5,2,4] => 19 [3,6,1,5,4,2] => 21 [3,6,2,1,4,5] => 16 [3,6,2,1,5,4] => 17 [3,6,2,4,1,5] => 19 [3,6,2,4,5,1] => 23 [3,6,2,5,1,4] => 21 [3,6,2,5,4,1] => 24 [3,6,4,1,2,5] => 20 [3,6,4,1,5,2] => 23 [3,6,4,2,1,5] => 21 [3,6,4,2,5,1] => 25 [3,6,4,5,1,2] => 27 [3,6,4,5,2,1] => 28 [3,6,5,1,2,4] => 23 [3,6,5,1,4,2] => 25 [3,6,5,2,1,4] => 24 [3,6,5,2,4,1] => 27 [3,6,5,4,1,2] => 28 [3,6,5,4,2,1] => 29 [4,1,2,3,5,6] => 6 [4,1,2,3,6,5] => 7 [4,1,2,5,3,6] => 8 [4,1,2,5,6,3] => 11 [4,1,2,6,3,5] => 10 [4,1,2,6,5,3] => 12 [4,1,3,2,5,6] => 7 [4,1,3,2,6,5] => 8 [4,1,3,5,2,6] => 10 [4,1,3,5,6,2] => 14 [4,1,3,6,2,5] => 12 [4,1,3,6,5,2] => 15 [4,1,5,2,3,6] => 11 [4,1,5,2,6,3] => 14 [4,1,5,3,2,6] => 12 [4,1,5,3,6,2] => 16 [4,1,5,6,2,3] => 18 [4,1,5,6,3,2] => 19 [4,1,6,2,3,5] => 14 [4,1,6,2,5,3] => 16 [4,1,6,3,2,5] => 15 [4,1,6,3,5,2] => 18 [4,1,6,5,2,3] => 19 [4,1,6,5,3,2] => 20 [4,2,1,3,5,6] => 7 [4,2,1,3,6,5] => 8 [4,2,1,5,3,6] => 9 [4,2,1,5,6,3] => 12 [4,2,1,6,3,5] => 11 [4,2,1,6,5,3] => 13 [4,2,3,1,5,6] => 9 [4,2,3,1,6,5] => 10 [4,2,3,5,1,6] => 13 [4,2,3,5,6,1] => 18 [4,2,3,6,1,5] => 15 [4,2,3,6,5,1] => 19 [4,2,5,1,3,6] => 13 [4,2,5,1,6,3] => 16 [4,2,5,3,1,6] => 15 [4,2,5,3,6,1] => 20 [4,2,5,6,1,3] => 21 [4,2,5,6,3,1] => 23 [4,2,6,1,3,5] => 16 [4,2,6,1,5,3] => 18 [4,2,6,3,1,5] => 18 [4,2,6,3,5,1] => 22 [4,2,6,5,1,3] => 22 [4,2,6,5,3,1] => 24 [4,3,1,2,5,6] => 9 [4,3,1,2,6,5] => 10 [4,3,1,5,2,6] => 12 [4,3,1,5,6,2] => 16 [4,3,1,6,2,5] => 14 [4,3,1,6,5,2] => 17 [4,3,2,1,5,6] => 10 [4,3,2,1,6,5] => 11 [4,3,2,5,1,6] => 14 [4,3,2,5,6,1] => 19 [4,3,2,6,1,5] => 16 [4,3,2,6,5,1] => 20 [4,3,5,1,2,6] => 16 [4,3,5,1,6,2] => 20 [4,3,5,2,1,6] => 17 [4,3,5,2,6,1] => 22 [4,3,5,6,1,2] => 25 [4,3,5,6,2,1] => 26 [4,3,6,1,2,5] => 19 [4,3,6,1,5,2] => 22 [4,3,6,2,1,5] => 20 [4,3,6,2,5,1] => 24 [4,3,6,5,1,2] => 26 [4,3,6,5,2,1] => 27 [4,5,1,2,3,6] => 15 [4,5,1,2,6,3] => 18 [4,5,1,3,2,6] => 16 [4,5,1,3,6,2] => 20 [4,5,1,6,2,3] => 22 [4,5,1,6,3,2] => 23 [4,5,2,1,3,6] => 16 [4,5,2,1,6,3] => 19 [4,5,2,3,1,6] => 18 [4,5,2,3,6,1] => 23 [4,5,2,6,1,3] => 24 [4,5,2,6,3,1] => 26 [4,5,3,1,2,6] => 18 [4,5,3,1,6,2] => 22 [4,5,3,2,1,6] => 19 [4,5,3,2,6,1] => 24 [4,5,3,6,1,2] => 27 [4,5,3,6,2,1] => 28 [4,5,6,1,2,3] => 27 [4,5,6,1,3,2] => 28 [4,5,6,2,1,3] => 28 [4,5,6,2,3,1] => 30 [4,5,6,3,1,2] => 30 [4,5,6,3,2,1] => 31 [4,6,1,2,3,5] => 19 [4,6,1,2,5,3] => 21 [4,6,1,3,2,5] => 20 [4,6,1,3,5,2] => 23 [4,6,1,5,2,3] => 24 [4,6,1,5,3,2] => 25 [4,6,2,1,3,5] => 20 [4,6,2,1,5,3] => 22 [4,6,2,3,1,5] => 22 [4,6,2,3,5,1] => 26 [4,6,2,5,1,3] => 26 [4,6,2,5,3,1] => 28 [4,6,3,1,2,5] => 22 [4,6,3,1,5,2] => 25 [4,6,3,2,1,5] => 23 [4,6,3,2,5,1] => 27 [4,6,3,5,1,2] => 29 [4,6,3,5,2,1] => 30 [4,6,5,1,2,3] => 28 [4,6,5,1,3,2] => 29 [4,6,5,2,1,3] => 29 [4,6,5,2,3,1] => 31 [4,6,5,3,1,2] => 31 [4,6,5,3,2,1] => 32 [5,1,2,3,4,6] => 10 [5,1,2,3,6,4] => 12 [5,1,2,4,3,6] => 11 [5,1,2,4,6,3] => 14 [5,1,2,6,3,4] => 15 [5,1,2,6,4,3] => 16 [5,1,3,2,4,6] => 11 [5,1,3,2,6,4] => 13 [5,1,3,4,2,6] => 13 [5,1,3,4,6,2] => 17 [5,1,3,6,2,4] => 17 [5,1,3,6,4,2] => 19 [5,1,4,2,3,6] => 13 [5,1,4,2,6,3] => 16 [5,1,4,3,2,6] => 14 [5,1,4,3,6,2] => 18 [5,1,4,6,2,3] => 20 [5,1,4,6,3,2] => 21 [5,1,6,2,3,4] => 19 [5,1,6,2,4,3] => 20 [5,1,6,3,2,4] => 20 [5,1,6,3,4,2] => 22 [5,1,6,4,2,3] => 22 [5,1,6,4,3,2] => 23 [5,2,1,3,4,6] => 11 [5,2,1,3,6,4] => 13 [5,2,1,4,3,6] => 12 [5,2,1,4,6,3] => 15 [5,2,1,6,3,4] => 16 [5,2,1,6,4,3] => 17 [5,2,3,1,4,6] => 13 [5,2,3,1,6,4] => 15 [5,2,3,4,1,6] => 16 [5,2,3,4,6,1] => 21 [5,2,3,6,1,4] => 20 [5,2,3,6,4,1] => 23 [5,2,4,1,3,6] => 15 [5,2,4,1,6,3] => 18 [5,2,4,3,1,6] => 17 [5,2,4,3,6,1] => 22 [5,2,4,6,1,3] => 23 [5,2,4,6,3,1] => 25 [5,2,6,1,3,4] => 21 [5,2,6,1,4,3] => 22 [5,2,6,3,1,4] => 23 [5,2,6,3,4,1] => 26 [5,2,6,4,1,3] => 25 [5,2,6,4,3,1] => 27 [5,3,1,2,4,6] => 13 [5,3,1,2,6,4] => 15 [5,3,1,4,2,6] => 15 [5,3,1,4,6,2] => 19 [5,3,1,6,2,4] => 19 [5,3,1,6,4,2] => 21 [5,3,2,1,4,6] => 14 [5,3,2,1,6,4] => 16 [5,3,2,4,1,6] => 17 [5,3,2,4,6,1] => 22 [5,3,2,6,1,4] => 21 [5,3,2,6,4,1] => 24 [5,3,4,1,2,6] => 18 [5,3,4,1,6,2] => 22 [5,3,4,2,1,6] => 19 [5,3,4,2,6,1] => 24 [5,3,4,6,1,2] => 27 [5,3,4,6,2,1] => 28 [5,3,6,1,2,4] => 24 [5,3,6,1,4,2] => 26 [5,3,6,2,1,4] => 25 [5,3,6,2,4,1] => 28 [5,3,6,4,1,2] => 29 [5,3,6,4,2,1] => 30 [5,4,1,2,3,6] => 16 [5,4,1,2,6,3] => 19 [5,4,1,3,2,6] => 17 [5,4,1,3,6,2] => 21 [5,4,1,6,2,3] => 23 [5,4,1,6,3,2] => 24 [5,4,2,1,3,6] => 17 [5,4,2,1,6,3] => 20 [5,4,2,3,1,6] => 19 [5,4,2,3,6,1] => 24 [5,4,2,6,1,3] => 25 [5,4,2,6,3,1] => 27 [5,4,3,1,2,6] => 19 [5,4,3,1,6,2] => 23 [5,4,3,2,1,6] => 20 [5,4,3,2,6,1] => 25 [5,4,3,6,1,2] => 28 [5,4,3,6,2,1] => 29 [5,4,6,1,2,3] => 28 [5,4,6,1,3,2] => 29 [5,4,6,2,1,3] => 29 [5,4,6,2,3,1] => 31 [5,4,6,3,1,2] => 31 [5,4,6,3,2,1] => 32 [5,6,1,2,3,4] => 24 [5,6,1,2,4,3] => 25 [5,6,1,3,2,4] => 25 [5,6,1,3,4,2] => 27 [5,6,1,4,2,3] => 27 [5,6,1,4,3,2] => 28 [5,6,2,1,3,4] => 25 [5,6,2,1,4,3] => 26 [5,6,2,3,1,4] => 27 [5,6,2,3,4,1] => 30 [5,6,2,4,1,3] => 29 [5,6,2,4,3,1] => 31 [5,6,3,1,2,4] => 27 [5,6,3,1,4,2] => 29 [5,6,3,2,1,4] => 28 [5,6,3,2,4,1] => 31 [5,6,3,4,1,2] => 32 [5,6,3,4,2,1] => 33 [5,6,4,1,2,3] => 30 [5,6,4,1,3,2] => 31 [5,6,4,2,1,3] => 31 [5,6,4,2,3,1] => 33 [5,6,4,3,1,2] => 33 [5,6,4,3,2,1] => 34 [6,1,2,3,4,5] => 15 [6,1,2,3,5,4] => 16 [6,1,2,4,3,5] => 16 [6,1,2,4,5,3] => 18 [6,1,2,5,3,4] => 18 [6,1,2,5,4,3] => 19 [6,1,3,2,4,5] => 16 [6,1,3,2,5,4] => 17 [6,1,3,4,2,5] => 18 [6,1,3,4,5,2] => 21 [6,1,3,5,2,4] => 20 [6,1,3,5,4,2] => 22 [6,1,4,2,3,5] => 18 [6,1,4,2,5,3] => 20 [6,1,4,3,2,5] => 19 [6,1,4,3,5,2] => 22 [6,1,4,5,2,3] => 23 [6,1,4,5,3,2] => 24 [6,1,5,2,3,4] => 21 [6,1,5,2,4,3] => 22 [6,1,5,3,2,4] => 22 [6,1,5,3,4,2] => 24 [6,1,5,4,2,3] => 24 [6,1,5,4,3,2] => 25 [6,2,1,3,4,5] => 16 [6,2,1,3,5,4] => 17 [6,2,1,4,3,5] => 17 [6,2,1,4,5,3] => 19 [6,2,1,5,3,4] => 19 [6,2,1,5,4,3] => 20 [6,2,3,1,4,5] => 18 [6,2,3,1,5,4] => 19 [6,2,3,4,1,5] => 21 [6,2,3,4,5,1] => 25 [6,2,3,5,1,4] => 23 [6,2,3,5,4,1] => 26 [6,2,4,1,3,5] => 20 [6,2,4,1,5,3] => 22 [6,2,4,3,1,5] => 22 [6,2,4,3,5,1] => 26 [6,2,4,5,1,3] => 26 [6,2,4,5,3,1] => 28 [6,2,5,1,3,4] => 23 [6,2,5,1,4,3] => 24 [6,2,5,3,1,4] => 25 [6,2,5,3,4,1] => 28 [6,2,5,4,1,3] => 27 [6,2,5,4,3,1] => 29 [6,3,1,2,4,5] => 18 [6,3,1,2,5,4] => 19 [6,3,1,4,2,5] => 20 [6,3,1,4,5,2] => 23 [6,3,1,5,2,4] => 22 [6,3,1,5,4,2] => 24 [6,3,2,1,4,5] => 19 [6,3,2,1,5,4] => 20 [6,3,2,4,1,5] => 22 [6,3,2,4,5,1] => 26 [6,3,2,5,1,4] => 24 [6,3,2,5,4,1] => 27 [6,3,4,1,2,5] => 23 [6,3,4,1,5,2] => 26 [6,3,4,2,1,5] => 24 [6,3,4,2,5,1] => 28 [6,3,4,5,1,2] => 30 [6,3,4,5,2,1] => 31 [6,3,5,1,2,4] => 26 [6,3,5,1,4,2] => 28 [6,3,5,2,1,4] => 27 [6,3,5,2,4,1] => 30 [6,3,5,4,1,2] => 31 [6,3,5,4,2,1] => 32 [6,4,1,2,3,5] => 21 [6,4,1,2,5,3] => 23 [6,4,1,3,2,5] => 22 [6,4,1,3,5,2] => 25 [6,4,1,5,2,3] => 26 [6,4,1,5,3,2] => 27 [6,4,2,1,3,5] => 22 [6,4,2,1,5,3] => 24 [6,4,2,3,1,5] => 24 [6,4,2,3,5,1] => 28 [6,4,2,5,1,3] => 28 [6,4,2,5,3,1] => 30 [6,4,3,1,2,5] => 24 [6,4,3,1,5,2] => 27 [6,4,3,2,1,5] => 25 [6,4,3,2,5,1] => 29 [6,4,3,5,1,2] => 31 [6,4,3,5,2,1] => 32 [6,4,5,1,2,3] => 30 [6,4,5,1,3,2] => 31 [6,4,5,2,1,3] => 31 [6,4,5,2,3,1] => 33 [6,4,5,3,1,2] => 33 [6,4,5,3,2,1] => 34 [6,5,1,2,3,4] => 25 [6,5,1,2,4,3] => 26 [6,5,1,3,2,4] => 26 [6,5,1,3,4,2] => 28 [6,5,1,4,2,3] => 28 [6,5,1,4,3,2] => 29 [6,5,2,1,3,4] => 26 [6,5,2,1,4,3] => 27 [6,5,2,3,1,4] => 28 [6,5,2,3,4,1] => 31 [6,5,2,4,1,3] => 30 [6,5,2,4,3,1] => 32 [6,5,3,1,2,4] => 28 [6,5,3,1,4,2] => 30 [6,5,3,2,1,4] => 29 [6,5,3,2,4,1] => 32 [6,5,3,4,1,2] => 33 [6,5,3,4,2,1] => 34 [6,5,4,1,2,3] => 31 [6,5,4,1,3,2] => 32 [6,5,4,2,1,3] => 32 [6,5,4,2,3,1] => 34 [6,5,4,3,1,2] => 34 [6,5,4,3,2,1] => 35 ----------------------------------------------------------------------------- Created: Mar 25, 2013 at 20:49 by Jessica Striker ----------------------------------------------------------------------------- Last Updated: May 30, 2019 at 11:28 by Masato Kobayashi