***************************************************************************** * 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: St000039 ----------------------------------------------------------------------------- Collection: Permutations ----------------------------------------------------------------------------- Description: The number of crossings of a permutation. A crossing of a permutation $\pi$ is given by a pair $(i,j)$ such that either $i < j \leq \pi(i) \leq \pi(j)$ or $\pi(i) < \pi(j) < i < j$. Pictorially, the diagram of a permutation is obtained by writing the numbers from $1$ to $n$ in this order on a line, and connecting $i$ and $\pi(i)$ with an arc above the line if $i\leq\pi(i)$ and with an arc below the line if $i > \pi(i)$. Then the number of crossings is the number of pairs of arcs above the line that cross or touch, plus the number of arcs below the line that cross. ----------------------------------------------------------------------------- References: [1] Corteel, S. Crossings and alignments of permutations [[MathSciNet:2290808]] ----------------------------------------------------------------------------- Code: def statistic(pi): n = len(pi) return sum( 1 for i in range(1,n+1) for j in range(1,n+1) if i < j <= pi(i) < pi(j) or pi(i) < pi(j) < i < j ) ----------------------------------------------------------------------------- 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] => 1 [3,1,2] => 0 [3,2,1] => 0 [1,2,3,4] => 0 [1,2,4,3] => 0 [1,3,2,4] => 0 [1,3,4,2] => 1 [1,4,2,3] => 0 [1,4,3,2] => 0 [2,1,3,4] => 0 [2,1,4,3] => 0 [2,3,1,4] => 1 [2,3,4,1] => 2 [2,4,1,3] => 1 [2,4,3,1] => 1 [3,1,2,4] => 0 [3,1,4,2] => 1 [3,2,1,4] => 0 [3,2,4,1] => 1 [3,4,1,2] => 2 [3,4,2,1] => 1 [4,1,2,3] => 0 [4,1,3,2] => 0 [4,2,1,3] => 0 [4,2,3,1] => 0 [4,3,1,2] => 1 [4,3,2,1] => 0 [1,2,3,4,5] => 0 [1,2,3,5,4] => 0 [1,2,4,3,5] => 0 [1,2,4,5,3] => 1 [1,2,5,3,4] => 0 [1,2,5,4,3] => 0 [1,3,2,4,5] => 0 [1,3,2,5,4] => 0 [1,3,4,2,5] => 1 [1,3,4,5,2] => 2 [1,3,5,2,4] => 1 [1,3,5,4,2] => 1 [1,4,2,3,5] => 0 [1,4,2,5,3] => 1 [1,4,3,2,5] => 0 [1,4,3,5,2] => 1 [1,4,5,2,3] => 2 [1,4,5,3,2] => 1 [1,5,2,3,4] => 0 [1,5,2,4,3] => 0 [1,5,3,2,4] => 0 [1,5,3,4,2] => 0 [1,5,4,2,3] => 1 [1,5,4,3,2] => 0 [2,1,3,4,5] => 0 [2,1,3,5,4] => 0 [2,1,4,3,5] => 0 [2,1,4,5,3] => 1 [2,1,5,3,4] => 0 [2,1,5,4,3] => 0 [2,3,1,4,5] => 1 [2,3,1,5,4] => 1 [2,3,4,1,5] => 2 [2,3,4,5,1] => 3 [2,3,5,1,4] => 2 [2,3,5,4,1] => 2 [2,4,1,3,5] => 1 [2,4,1,5,3] => 2 [2,4,3,1,5] => 1 [2,4,3,5,1] => 2 [2,4,5,1,3] => 3 [2,4,5,3,1] => 2 [2,5,1,3,4] => 1 [2,5,1,4,3] => 1 [2,5,3,1,4] => 1 [2,5,3,4,1] => 1 [2,5,4,1,3] => 2 [2,5,4,3,1] => 1 [3,1,2,4,5] => 0 [3,1,2,5,4] => 0 [3,1,4,2,5] => 1 [3,1,4,5,2] => 2 [3,1,5,2,4] => 1 [3,1,5,4,2] => 1 [3,2,1,4,5] => 0 [3,2,1,5,4] => 0 [3,2,4,1,5] => 1 [3,2,4,5,1] => 2 [3,2,5,1,4] => 1 [3,2,5,4,1] => 1 [3,4,1,2,5] => 2 [3,4,1,5,2] => 3 [3,4,2,1,5] => 1 [3,4,2,5,1] => 2 [3,4,5,1,2] => 4 [3,4,5,2,1] => 3 [3,5,1,2,4] => 2 [3,5,1,4,2] => 2 [3,5,2,1,4] => 1 [3,5,2,4,1] => 1 [3,5,4,1,2] => 3 [3,5,4,2,1] => 2 [4,1,2,3,5] => 0 [4,1,2,5,3] => 1 [4,1,3,2,5] => 0 [4,1,3,5,2] => 1 [4,1,5,2,3] => 2 [4,1,5,3,2] => 1 [4,2,1,3,5] => 0 [4,2,1,5,3] => 1 [4,2,3,1,5] => 0 [4,2,3,5,1] => 1 [4,2,5,1,3] => 2 [4,2,5,3,1] => 1 [4,3,1,2,5] => 1 [4,3,1,5,2] => 2 [4,3,2,1,5] => 0 [4,3,2,5,1] => 1 [4,3,5,1,2] => 3 [4,3,5,2,1] => 2 [4,5,1,2,3] => 3 [4,5,1,3,2] => 2 [4,5,2,1,3] => 2 [4,5,2,3,1] => 1 [4,5,3,1,2] => 2 [4,5,3,2,1] => 1 [5,1,2,3,4] => 0 [5,1,2,4,3] => 0 [5,1,3,2,4] => 0 [5,1,3,4,2] => 0 [5,1,4,2,3] => 1 [5,1,4,3,2] => 0 [5,2,1,3,4] => 0 [5,2,1,4,3] => 0 [5,2,3,1,4] => 0 [5,2,3,4,1] => 0 [5,2,4,1,3] => 1 [5,2,4,3,1] => 0 [5,3,1,2,4] => 1 [5,3,1,4,2] => 1 [5,3,2,1,4] => 0 [5,3,2,4,1] => 0 [5,3,4,1,2] => 2 [5,3,4,2,1] => 1 [5,4,1,2,3] => 2 [5,4,1,3,2] => 1 [5,4,2,1,3] => 1 [5,4,2,3,1] => 0 [5,4,3,1,2] => 1 [5,4,3,2,1] => 0 [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] => 1 [1,2,3,6,4,5] => 0 [1,2,3,6,5,4] => 0 [1,2,4,3,5,6] => 0 [1,2,4,3,6,5] => 0 [1,2,4,5,3,6] => 1 [1,2,4,5,6,3] => 2 [1,2,4,6,3,5] => 1 [1,2,4,6,5,3] => 1 [1,2,5,3,4,6] => 0 [1,2,5,3,6,4] => 1 [1,2,5,4,3,6] => 0 [1,2,5,4,6,3] => 1 [1,2,5,6,3,4] => 2 [1,2,5,6,4,3] => 1 [1,2,6,3,4,5] => 0 [1,2,6,3,5,4] => 0 [1,2,6,4,3,5] => 0 [1,2,6,4,5,3] => 0 [1,2,6,5,3,4] => 1 [1,2,6,5,4,3] => 0 [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] => 1 [1,3,2,6,4,5] => 0 [1,3,2,6,5,4] => 0 [1,3,4,2,5,6] => 1 [1,3,4,2,6,5] => 1 [1,3,4,5,2,6] => 2 [1,3,4,5,6,2] => 3 [1,3,4,6,2,5] => 2 [1,3,4,6,5,2] => 2 [1,3,5,2,4,6] => 1 [1,3,5,2,6,4] => 2 [1,3,5,4,2,6] => 1 [1,3,5,4,6,2] => 2 [1,3,5,6,2,4] => 3 [1,3,5,6,4,2] => 2 [1,3,6,2,4,5] => 1 [1,3,6,2,5,4] => 1 [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] => 1 [1,4,2,3,5,6] => 0 [1,4,2,3,6,5] => 0 [1,4,2,5,3,6] => 1 [1,4,2,5,6,3] => 2 [1,4,2,6,3,5] => 1 [1,4,2,6,5,3] => 1 [1,4,3,2,5,6] => 0 [1,4,3,2,6,5] => 0 [1,4,3,5,2,6] => 1 [1,4,3,5,6,2] => 2 [1,4,3,6,2,5] => 1 [1,4,3,6,5,2] => 1 [1,4,5,2,3,6] => 2 [1,4,5,2,6,3] => 3 [1,4,5,3,2,6] => 1 [1,4,5,3,6,2] => 2 [1,4,5,6,2,3] => 4 [1,4,5,6,3,2] => 3 [1,4,6,2,3,5] => 2 [1,4,6,2,5,3] => 2 [1,4,6,3,2,5] => 1 [1,4,6,3,5,2] => 1 [1,4,6,5,2,3] => 3 [1,4,6,5,3,2] => 2 [1,5,2,3,4,6] => 0 [1,5,2,3,6,4] => 1 [1,5,2,4,3,6] => 0 [1,5,2,4,6,3] => 1 [1,5,2,6,3,4] => 2 [1,5,2,6,4,3] => 1 [1,5,3,2,4,6] => 0 [1,5,3,2,6,4] => 1 [1,5,3,4,2,6] => 0 [1,5,3,4,6,2] => 1 [1,5,3,6,2,4] => 2 [1,5,3,6,4,2] => 1 [1,5,4,2,3,6] => 1 [1,5,4,2,6,3] => 2 [1,5,4,3,2,6] => 0 [1,5,4,3,6,2] => 1 [1,5,4,6,2,3] => 3 [1,5,4,6,3,2] => 2 [1,5,6,2,3,4] => 3 [1,5,6,2,4,3] => 2 [1,5,6,3,2,4] => 2 [1,5,6,3,4,2] => 1 [1,5,6,4,2,3] => 2 [1,5,6,4,3,2] => 1 [1,6,2,3,4,5] => 0 [1,6,2,3,5,4] => 0 [1,6,2,4,3,5] => 0 [1,6,2,4,5,3] => 0 [1,6,2,5,3,4] => 1 [1,6,2,5,4,3] => 0 [1,6,3,2,4,5] => 0 [1,6,3,2,5,4] => 0 [1,6,3,4,2,5] => 0 [1,6,3,4,5,2] => 0 [1,6,3,5,2,4] => 1 [1,6,3,5,4,2] => 0 [1,6,4,2,3,5] => 1 [1,6,4,2,5,3] => 1 [1,6,4,3,2,5] => 0 [1,6,4,3,5,2] => 0 [1,6,4,5,2,3] => 2 [1,6,4,5,3,2] => 1 [1,6,5,2,3,4] => 2 [1,6,5,2,4,3] => 1 [1,6,5,3,2,4] => 1 [1,6,5,3,4,2] => 0 [1,6,5,4,2,3] => 1 [1,6,5,4,3,2] => 0 [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] => 1 [2,1,3,6,4,5] => 0 [2,1,3,6,5,4] => 0 [2,1,4,3,5,6] => 0 [2,1,4,3,6,5] => 0 [2,1,4,5,3,6] => 1 [2,1,4,5,6,3] => 2 [2,1,4,6,3,5] => 1 [2,1,4,6,5,3] => 1 [2,1,5,3,4,6] => 0 [2,1,5,3,6,4] => 1 [2,1,5,4,3,6] => 0 [2,1,5,4,6,3] => 1 [2,1,5,6,3,4] => 2 [2,1,5,6,4,3] => 1 [2,1,6,3,4,5] => 0 [2,1,6,3,5,4] => 0 [2,1,6,4,3,5] => 0 [2,1,6,4,5,3] => 0 [2,1,6,5,3,4] => 1 [2,1,6,5,4,3] => 0 [2,3,1,4,5,6] => 1 [2,3,1,4,6,5] => 1 [2,3,1,5,4,6] => 1 [2,3,1,5,6,4] => 2 [2,3,1,6,4,5] => 1 [2,3,1,6,5,4] => 1 [2,3,4,1,5,6] => 2 [2,3,4,1,6,5] => 2 [2,3,4,5,1,6] => 3 [2,3,4,5,6,1] => 4 [2,3,4,6,1,5] => 3 [2,3,4,6,5,1] => 3 [2,3,5,1,4,6] => 2 [2,3,5,1,6,4] => 3 [2,3,5,4,1,6] => 2 [2,3,5,4,6,1] => 3 [2,3,5,6,1,4] => 4 [2,3,5,6,4,1] => 3 [2,3,6,1,4,5] => 2 [2,3,6,1,5,4] => 2 [2,3,6,4,1,5] => 2 [2,3,6,4,5,1] => 2 [2,3,6,5,1,4] => 3 [2,3,6,5,4,1] => 2 [2,4,1,3,5,6] => 1 [2,4,1,3,6,5] => 1 [2,4,1,5,3,6] => 2 [2,4,1,5,6,3] => 3 [2,4,1,6,3,5] => 2 [2,4,1,6,5,3] => 2 [2,4,3,1,5,6] => 1 [2,4,3,1,6,5] => 1 [2,4,3,5,1,6] => 2 [2,4,3,5,6,1] => 3 [2,4,3,6,1,5] => 2 [2,4,3,6,5,1] => 2 [2,4,5,1,3,6] => 3 [2,4,5,1,6,3] => 4 [2,4,5,3,1,6] => 2 [2,4,5,3,6,1] => 3 [2,4,5,6,1,3] => 5 [2,4,5,6,3,1] => 4 [2,4,6,1,3,5] => 3 [2,4,6,1,5,3] => 3 [2,4,6,3,1,5] => 2 [2,4,6,3,5,1] => 2 [2,4,6,5,1,3] => 4 [2,4,6,5,3,1] => 3 [2,5,1,3,4,6] => 1 [2,5,1,3,6,4] => 2 [2,5,1,4,3,6] => 1 [2,5,1,4,6,3] => 2 [2,5,1,6,3,4] => 3 [2,5,1,6,4,3] => 2 [2,5,3,1,4,6] => 1 [2,5,3,1,6,4] => 2 [2,5,3,4,1,6] => 1 [2,5,3,4,6,1] => 2 [2,5,3,6,1,4] => 3 [2,5,3,6,4,1] => 2 [2,5,4,1,3,6] => 2 [2,5,4,1,6,3] => 3 [2,5,4,3,1,6] => 1 [2,5,4,3,6,1] => 2 [2,5,4,6,1,3] => 4 [2,5,4,6,3,1] => 3 [2,5,6,1,3,4] => 4 [2,5,6,1,4,3] => 3 [2,5,6,3,1,4] => 3 [2,5,6,3,4,1] => 2 [2,5,6,4,1,3] => 3 [2,5,6,4,3,1] => 2 [2,6,1,3,4,5] => 1 [2,6,1,3,5,4] => 1 [2,6,1,4,3,5] => 1 [2,6,1,4,5,3] => 1 [2,6,1,5,3,4] => 2 [2,6,1,5,4,3] => 1 [2,6,3,1,4,5] => 1 [2,6,3,1,5,4] => 1 [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] => 1 [2,6,4,1,3,5] => 2 [2,6,4,1,5,3] => 2 [2,6,4,3,1,5] => 1 [2,6,4,3,5,1] => 1 [2,6,4,5,1,3] => 3 [2,6,4,5,3,1] => 2 [2,6,5,1,3,4] => 3 [2,6,5,1,4,3] => 2 [2,6,5,3,1,4] => 2 [2,6,5,3,4,1] => 1 [2,6,5,4,1,3] => 2 [2,6,5,4,3,1] => 1 [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] => 1 [3,1,2,6,4,5] => 0 [3,1,2,6,5,4] => 0 [3,1,4,2,5,6] => 1 [3,1,4,2,6,5] => 1 [3,1,4,5,2,6] => 2 [3,1,4,5,6,2] => 3 [3,1,4,6,2,5] => 2 [3,1,4,6,5,2] => 2 [3,1,5,2,4,6] => 1 [3,1,5,2,6,4] => 2 [3,1,5,4,2,6] => 1 [3,1,5,4,6,2] => 2 [3,1,5,6,2,4] => 3 [3,1,5,6,4,2] => 2 [3,1,6,2,4,5] => 1 [3,1,6,2,5,4] => 1 [3,1,6,4,2,5] => 1 [3,1,6,4,5,2] => 1 [3,1,6,5,2,4] => 2 [3,1,6,5,4,2] => 1 [3,2,1,4,5,6] => 0 [3,2,1,4,6,5] => 0 [3,2,1,5,4,6] => 0 [3,2,1,5,6,4] => 1 [3,2,1,6,4,5] => 0 [3,2,1,6,5,4] => 0 [3,2,4,1,5,6] => 1 [3,2,4,1,6,5] => 1 [3,2,4,5,1,6] => 2 [3,2,4,5,6,1] => 3 [3,2,4,6,1,5] => 2 [3,2,4,6,5,1] => 2 [3,2,5,1,4,6] => 1 [3,2,5,1,6,4] => 2 [3,2,5,4,1,6] => 1 [3,2,5,4,6,1] => 2 [3,2,5,6,1,4] => 3 [3,2,5,6,4,1] => 2 [3,2,6,1,4,5] => 1 [3,2,6,1,5,4] => 1 [3,2,6,4,1,5] => 1 [3,2,6,4,5,1] => 1 [3,2,6,5,1,4] => 2 [3,2,6,5,4,1] => 1 [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] => 4 [3,4,1,6,2,5] => 3 [3,4,1,6,5,2] => 3 [3,4,2,1,5,6] => 1 [3,4,2,1,6,5] => 1 [3,4,2,5,1,6] => 2 [3,4,2,5,6,1] => 3 [3,4,2,6,1,5] => 2 [3,4,2,6,5,1] => 2 [3,4,5,1,2,6] => 4 [3,4,5,1,6,2] => 5 [3,4,5,2,1,6] => 3 [3,4,5,2,6,1] => 4 [3,4,5,6,1,2] => 6 [3,4,5,6,2,1] => 5 [3,4,6,1,2,5] => 4 [3,4,6,1,5,2] => 4 [3,4,6,2,1,5] => 3 [3,4,6,2,5,1] => 3 [3,4,6,5,1,2] => 5 [3,4,6,5,2,1] => 4 [3,5,1,2,4,6] => 2 [3,5,1,2,6,4] => 3 [3,5,1,4,2,6] => 2 [3,5,1,4,6,2] => 3 [3,5,1,6,2,4] => 4 [3,5,1,6,4,2] => 3 [3,5,2,1,4,6] => 1 [3,5,2,1,6,4] => 2 [3,5,2,4,1,6] => 1 [3,5,2,4,6,1] => 2 [3,5,2,6,1,4] => 3 [3,5,2,6,4,1] => 2 [3,5,4,1,2,6] => 3 [3,5,4,1,6,2] => 4 [3,5,4,2,1,6] => 2 [3,5,4,2,6,1] => 3 [3,5,4,6,1,2] => 5 [3,5,4,6,2,1] => 4 [3,5,6,1,2,4] => 5 [3,5,6,1,4,2] => 4 [3,5,6,2,1,4] => 4 [3,5,6,2,4,1] => 3 [3,5,6,4,1,2] => 4 [3,5,6,4,2,1] => 3 [3,6,1,2,4,5] => 2 [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] => 3 [3,6,1,5,4,2] => 2 [3,6,2,1,4,5] => 1 [3,6,2,1,5,4] => 1 [3,6,2,4,1,5] => 1 [3,6,2,4,5,1] => 1 [3,6,2,5,1,4] => 2 [3,6,2,5,4,1] => 1 [3,6,4,1,2,5] => 3 [3,6,4,1,5,2] => 3 [3,6,4,2,1,5] => 2 [3,6,4,2,5,1] => 2 [3,6,4,5,1,2] => 4 [3,6,4,5,2,1] => 3 [3,6,5,1,2,4] => 4 [3,6,5,1,4,2] => 3 [3,6,5,2,1,4] => 3 [3,6,5,2,4,1] => 2 [3,6,5,4,1,2] => 3 [3,6,5,4,2,1] => 2 [4,1,2,3,5,6] => 0 [4,1,2,3,6,5] => 0 [4,1,2,5,3,6] => 1 [4,1,2,5,6,3] => 2 [4,1,2,6,3,5] => 1 [4,1,2,6,5,3] => 1 [4,1,3,2,5,6] => 0 [4,1,3,2,6,5] => 0 [4,1,3,5,2,6] => 1 [4,1,3,5,6,2] => 2 [4,1,3,6,2,5] => 1 [4,1,3,6,5,2] => 1 [4,1,5,2,3,6] => 2 [4,1,5,2,6,3] => 3 [4,1,5,3,2,6] => 1 [4,1,5,3,6,2] => 2 [4,1,5,6,2,3] => 4 [4,1,5,6,3,2] => 3 [4,1,6,2,3,5] => 2 [4,1,6,2,5,3] => 2 [4,1,6,3,2,5] => 1 [4,1,6,3,5,2] => 1 [4,1,6,5,2,3] => 3 [4,1,6,5,3,2] => 2 [4,2,1,3,5,6] => 0 [4,2,1,3,6,5] => 0 [4,2,1,5,3,6] => 1 [4,2,1,5,6,3] => 2 [4,2,1,6,3,5] => 1 [4,2,1,6,5,3] => 1 [4,2,3,1,5,6] => 0 [4,2,3,1,6,5] => 0 [4,2,3,5,1,6] => 1 [4,2,3,5,6,1] => 2 [4,2,3,6,1,5] => 1 [4,2,3,6,5,1] => 1 [4,2,5,1,3,6] => 2 [4,2,5,1,6,3] => 3 [4,2,5,3,1,6] => 1 [4,2,5,3,6,1] => 2 [4,2,5,6,1,3] => 4 [4,2,5,6,3,1] => 3 [4,2,6,1,3,5] => 2 [4,2,6,1,5,3] => 2 [4,2,6,3,1,5] => 1 [4,2,6,3,5,1] => 1 [4,2,6,5,1,3] => 3 [4,2,6,5,3,1] => 2 [4,3,1,2,5,6] => 1 [4,3,1,2,6,5] => 1 [4,3,1,5,2,6] => 2 [4,3,1,5,6,2] => 3 [4,3,1,6,2,5] => 2 [4,3,1,6,5,2] => 2 [4,3,2,1,5,6] => 0 [4,3,2,1,6,5] => 0 [4,3,2,5,1,6] => 1 [4,3,2,5,6,1] => 2 [4,3,2,6,1,5] => 1 [4,3,2,6,5,1] => 1 [4,3,5,1,2,6] => 3 [4,3,5,1,6,2] => 4 [4,3,5,2,1,6] => 2 [4,3,5,2,6,1] => 3 [4,3,5,6,1,2] => 5 [4,3,5,6,2,1] => 4 [4,3,6,1,2,5] => 3 [4,3,6,1,5,2] => 3 [4,3,6,2,1,5] => 2 [4,3,6,2,5,1] => 2 [4,3,6,5,1,2] => 4 [4,3,6,5,2,1] => 3 [4,5,1,2,3,6] => 3 [4,5,1,2,6,3] => 4 [4,5,1,3,2,6] => 2 [4,5,1,3,6,2] => 3 [4,5,1,6,2,3] => 5 [4,5,1,6,3,2] => 4 [4,5,2,1,3,6] => 2 [4,5,2,1,6,3] => 3 [4,5,2,3,1,6] => 1 [4,5,2,3,6,1] => 2 [4,5,2,6,1,3] => 4 [4,5,2,6,3,1] => 3 [4,5,3,1,2,6] => 2 [4,5,3,1,6,2] => 3 [4,5,3,2,1,6] => 1 [4,5,3,2,6,1] => 2 [4,5,3,6,1,2] => 4 [4,5,3,6,2,1] => 3 [4,5,6,1,2,3] => 6 [4,5,6,1,3,2] => 5 [4,5,6,2,1,3] => 5 [4,5,6,2,3,1] => 4 [4,5,6,3,1,2] => 4 [4,5,6,3,2,1] => 3 [4,6,1,2,3,5] => 3 [4,6,1,2,5,3] => 3 [4,6,1,3,2,5] => 2 [4,6,1,3,5,2] => 2 [4,6,1,5,2,3] => 4 [4,6,1,5,3,2] => 3 [4,6,2,1,3,5] => 2 [4,6,2,1,5,3] => 2 [4,6,2,3,1,5] => 1 [4,6,2,3,5,1] => 1 [4,6,2,5,1,3] => 3 [4,6,2,5,3,1] => 2 [4,6,3,1,2,5] => 2 [4,6,3,1,5,2] => 2 [4,6,3,2,1,5] => 1 [4,6,3,2,5,1] => 1 [4,6,3,5,1,2] => 3 [4,6,3,5,2,1] => 2 [4,6,5,1,2,3] => 5 [4,6,5,1,3,2] => 4 [4,6,5,2,1,3] => 4 [4,6,5,2,3,1] => 3 [4,6,5,3,1,2] => 3 [4,6,5,3,2,1] => 2 [5,1,2,3,4,6] => 0 [5,1,2,3,6,4] => 1 [5,1,2,4,3,6] => 0 [5,1,2,4,6,3] => 1 [5,1,2,6,3,4] => 2 [5,1,2,6,4,3] => 1 [5,1,3,2,4,6] => 0 [5,1,3,2,6,4] => 1 [5,1,3,4,2,6] => 0 [5,1,3,4,6,2] => 1 [5,1,3,6,2,4] => 2 [5,1,3,6,4,2] => 1 [5,1,4,2,3,6] => 1 [5,1,4,2,6,3] => 2 [5,1,4,3,2,6] => 0 [5,1,4,3,6,2] => 1 [5,1,4,6,2,3] => 3 [5,1,4,6,3,2] => 2 [5,1,6,2,3,4] => 3 [5,1,6,2,4,3] => 2 [5,1,6,3,2,4] => 2 [5,1,6,3,4,2] => 1 [5,1,6,4,2,3] => 2 [5,1,6,4,3,2] => 1 [5,2,1,3,4,6] => 0 [5,2,1,3,6,4] => 1 [5,2,1,4,3,6] => 0 [5,2,1,4,6,3] => 1 [5,2,1,6,3,4] => 2 [5,2,1,6,4,3] => 1 [5,2,3,1,4,6] => 0 [5,2,3,1,6,4] => 1 [5,2,3,4,1,6] => 0 [5,2,3,4,6,1] => 1 [5,2,3,6,1,4] => 2 [5,2,3,6,4,1] => 1 [5,2,4,1,3,6] => 1 [5,2,4,1,6,3] => 2 [5,2,4,3,1,6] => 0 [5,2,4,3,6,1] => 1 [5,2,4,6,1,3] => 3 [5,2,4,6,3,1] => 2 [5,2,6,1,3,4] => 3 [5,2,6,1,4,3] => 2 [5,2,6,3,1,4] => 2 [5,2,6,3,4,1] => 1 [5,2,6,4,1,3] => 2 [5,2,6,4,3,1] => 1 [5,3,1,2,4,6] => 1 [5,3,1,2,6,4] => 2 [5,3,1,4,2,6] => 1 [5,3,1,4,6,2] => 2 [5,3,1,6,2,4] => 3 [5,3,1,6,4,2] => 2 [5,3,2,1,4,6] => 0 [5,3,2,1,6,4] => 1 [5,3,2,4,1,6] => 0 [5,3,2,4,6,1] => 1 [5,3,2,6,1,4] => 2 [5,3,2,6,4,1] => 1 [5,3,4,1,2,6] => 2 [5,3,4,1,6,2] => 3 [5,3,4,2,1,6] => 1 [5,3,4,2,6,1] => 2 [5,3,4,6,1,2] => 4 [5,3,4,6,2,1] => 3 [5,3,6,1,2,4] => 4 [5,3,6,1,4,2] => 3 [5,3,6,2,1,4] => 3 [5,3,6,2,4,1] => 2 [5,3,6,4,1,2] => 3 [5,3,6,4,2,1] => 2 [5,4,1,2,3,6] => 2 [5,4,1,2,6,3] => 3 [5,4,1,3,2,6] => 1 [5,4,1,3,6,2] => 2 [5,4,1,6,2,3] => 4 [5,4,1,6,3,2] => 3 [5,4,2,1,3,6] => 1 [5,4,2,1,6,3] => 2 [5,4,2,3,1,6] => 0 [5,4,2,3,6,1] => 1 [5,4,2,6,1,3] => 3 [5,4,2,6,3,1] => 2 [5,4,3,1,2,6] => 1 [5,4,3,1,6,2] => 2 [5,4,3,2,1,6] => 0 [5,4,3,2,6,1] => 1 [5,4,3,6,1,2] => 3 [5,4,3,6,2,1] => 2 [5,4,6,1,2,3] => 5 [5,4,6,1,3,2] => 4 [5,4,6,2,1,3] => 4 [5,4,6,2,3,1] => 3 [5,4,6,3,1,2] => 3 [5,4,6,3,2,1] => 2 [5,6,1,2,3,4] => 4 [5,6,1,2,4,3] => 3 [5,6,1,3,2,4] => 3 [5,6,1,3,4,2] => 2 [5,6,1,4,2,3] => 3 [5,6,1,4,3,2] => 2 [5,6,2,1,3,4] => 3 [5,6,2,1,4,3] => 2 [5,6,2,3,1,4] => 2 [5,6,2,3,4,1] => 1 [5,6,2,4,1,3] => 2 [5,6,2,4,3,1] => 1 [5,6,3,1,2,4] => 3 [5,6,3,1,4,2] => 2 [5,6,3,2,1,4] => 2 [5,6,3,2,4,1] => 1 [5,6,3,4,1,2] => 2 [5,6,3,4,2,1] => 1 [5,6,4,1,2,3] => 4 [5,6,4,1,3,2] => 3 [5,6,4,2,1,3] => 3 [5,6,4,2,3,1] => 2 [5,6,4,3,1,2] => 2 [5,6,4,3,2,1] => 1 [6,1,2,3,4,5] => 0 [6,1,2,3,5,4] => 0 [6,1,2,4,3,5] => 0 [6,1,2,4,5,3] => 0 [6,1,2,5,3,4] => 1 [6,1,2,5,4,3] => 0 [6,1,3,2,4,5] => 0 [6,1,3,2,5,4] => 0 [6,1,3,4,2,5] => 0 [6,1,3,4,5,2] => 0 [6,1,3,5,2,4] => 1 [6,1,3,5,4,2] => 0 [6,1,4,2,3,5] => 1 [6,1,4,2,5,3] => 1 [6,1,4,3,2,5] => 0 [6,1,4,3,5,2] => 0 [6,1,4,5,2,3] => 2 [6,1,4,5,3,2] => 1 [6,1,5,2,3,4] => 2 [6,1,5,2,4,3] => 1 [6,1,5,3,2,4] => 1 [6,1,5,3,4,2] => 0 [6,1,5,4,2,3] => 1 [6,1,5,4,3,2] => 0 [6,2,1,3,4,5] => 0 [6,2,1,3,5,4] => 0 [6,2,1,4,3,5] => 0 [6,2,1,4,5,3] => 0 [6,2,1,5,3,4] => 1 [6,2,1,5,4,3] => 0 [6,2,3,1,4,5] => 0 [6,2,3,1,5,4] => 0 [6,2,3,4,1,5] => 0 [6,2,3,4,5,1] => 0 [6,2,3,5,1,4] => 1 [6,2,3,5,4,1] => 0 [6,2,4,1,3,5] => 1 [6,2,4,1,5,3] => 1 [6,2,4,3,1,5] => 0 [6,2,4,3,5,1] => 0 [6,2,4,5,1,3] => 2 [6,2,4,5,3,1] => 1 [6,2,5,1,3,4] => 2 [6,2,5,1,4,3] => 1 [6,2,5,3,1,4] => 1 [6,2,5,3,4,1] => 0 [6,2,5,4,1,3] => 1 [6,2,5,4,3,1] => 0 [6,3,1,2,4,5] => 1 [6,3,1,2,5,4] => 1 [6,3,1,4,2,5] => 1 [6,3,1,4,5,2] => 1 [6,3,1,5,2,4] => 2 [6,3,1,5,4,2] => 1 [6,3,2,1,4,5] => 0 [6,3,2,1,5,4] => 0 [6,3,2,4,1,5] => 0 [6,3,2,4,5,1] => 0 [6,3,2,5,1,4] => 1 [6,3,2,5,4,1] => 0 [6,3,4,1,2,5] => 2 [6,3,4,1,5,2] => 2 [6,3,4,2,1,5] => 1 [6,3,4,2,5,1] => 1 [6,3,4,5,1,2] => 3 [6,3,4,5,2,1] => 2 [6,3,5,1,2,4] => 3 [6,3,5,1,4,2] => 2 [6,3,5,2,1,4] => 2 [6,3,5,2,4,1] => 1 [6,3,5,4,1,2] => 2 [6,3,5,4,2,1] => 1 [6,4,1,2,3,5] => 2 [6,4,1,2,5,3] => 2 [6,4,1,3,2,5] => 1 [6,4,1,3,5,2] => 1 [6,4,1,5,2,3] => 3 [6,4,1,5,3,2] => 2 [6,4,2,1,3,5] => 1 [6,4,2,1,5,3] => 1 [6,4,2,3,1,5] => 0 [6,4,2,3,5,1] => 0 [6,4,2,5,1,3] => 2 [6,4,2,5,3,1] => 1 [6,4,3,1,2,5] => 1 [6,4,3,1,5,2] => 1 [6,4,3,2,1,5] => 0 [6,4,3,2,5,1] => 0 [6,4,3,5,1,2] => 2 [6,4,3,5,2,1] => 1 [6,4,5,1,2,3] => 4 [6,4,5,1,3,2] => 3 [6,4,5,2,1,3] => 3 [6,4,5,2,3,1] => 2 [6,4,5,3,1,2] => 2 [6,4,5,3,2,1] => 1 [6,5,1,2,3,4] => 3 [6,5,1,2,4,3] => 2 [6,5,1,3,2,4] => 2 [6,5,1,3,4,2] => 1 [6,5,1,4,2,3] => 2 [6,5,1,4,3,2] => 1 [6,5,2,1,3,4] => 2 [6,5,2,1,4,3] => 1 [6,5,2,3,1,4] => 1 [6,5,2,3,4,1] => 0 [6,5,2,4,1,3] => 1 [6,5,2,4,3,1] => 0 [6,5,3,1,2,4] => 2 [6,5,3,1,4,2] => 1 [6,5,3,2,1,4] => 1 [6,5,3,2,4,1] => 0 [6,5,3,4,1,2] => 1 [6,5,3,4,2,1] => 0 [6,5,4,1,2,3] => 3 [6,5,4,1,3,2] => 2 [6,5,4,2,1,3] => 2 [6,5,4,2,3,1] => 1 [6,5,4,3,1,2] => 1 [6,5,4,3,2,1] => 0 ----------------------------------------------------------------------------- Created: Feb 21, 2013 at 01:01 by Mirkó Visontai ----------------------------------------------------------------------------- Last Updated: May 10, 2019 at 17:13 by Henning Ulfarsson