***************************************************************************** * 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: St001220 ----------------------------------------------------------------------------- Collection: Permutations ----------------------------------------------------------------------------- Description: The width of a permutation. Let $w$ be a permutation. The interval $[e,w]$ in the weak order is ranked, and we define $r_i=r_i(w)$ to be the number of elements at rank $i$ in $[e,w]$, where $i \in \{0, \dots, \ell(w)\}$. The ''width'' of $w$ is the maximum of $\{r_0,r_1,\ldots,r_{\ell(w)}\}$. See [1]. ----------------------------------------------------------------------------- References: [1] Fishel, S., MiliÄ‡eviÄ‡, E., Patrias, R., Tenner, B. E. Enumerations relating braid and commutation classes [[arXiv:1708.04372]] ----------------------------------------------------------------------------- Code: def statistic(pi): S = SymmetricGroup(len(pi)) pi = S(pi) P = S.weak_poset() return P.subposet(P.principal_order_ideal(pi)).width() ----------------------------------------------------------------------------- Statistic values: [1,2] => 1 [2,1] => 1 [1,2,3] => 1 [1,3,2] => 1 [2,1,3] => 1 [2,3,1] => 1 [3,1,2] => 1 [3,2,1] => 2 [1,2,3,4] => 1 [1,2,4,3] => 1 [1,3,2,4] => 1 [1,3,4,2] => 1 [1,4,2,3] => 1 [1,4,3,2] => 2 [2,1,3,4] => 1 [2,1,4,3] => 2 [2,3,1,4] => 1 [2,3,4,1] => 1 [2,4,1,3] => 2 [2,4,3,1] => 2 [3,1,2,4] => 1 [3,1,4,2] => 2 [3,2,1,4] => 2 [3,2,4,1] => 2 [3,4,1,2] => 2 [3,4,2,1] => 3 [4,1,2,3] => 1 [4,1,3,2] => 2 [4,2,1,3] => 2 [4,2,3,1] => 3 [4,3,1,2] => 3 [4,3,2,1] => 6 [1,2,3,4,5] => 1 [1,2,3,5,4] => 1 [1,2,4,3,5] => 1 [1,2,4,5,3] => 1 [1,2,5,3,4] => 1 [1,2,5,4,3] => 2 [1,3,2,4,5] => 1 [1,3,2,5,4] => 2 [1,3,4,2,5] => 1 [1,3,4,5,2] => 1 [1,3,5,2,4] => 2 [1,3,5,4,2] => 2 [1,4,2,3,5] => 1 [1,4,2,5,3] => 2 [1,4,3,2,5] => 2 [1,4,3,5,2] => 2 [1,4,5,2,3] => 2 [1,4,5,3,2] => 3 [1,5,2,3,4] => 1 [1,5,2,4,3] => 2 [1,5,3,2,4] => 2 [1,5,3,4,2] => 3 [1,5,4,2,3] => 3 [1,5,4,3,2] => 6 [2,1,3,4,5] => 1 [2,1,3,5,4] => 2 [2,1,4,3,5] => 2 [2,1,4,5,3] => 2 [2,1,5,3,4] => 2 [2,1,5,4,3] => 4 [2,3,1,4,5] => 1 [2,3,1,5,4] => 2 [2,3,4,1,5] => 1 [2,3,4,5,1] => 1 [2,3,5,1,4] => 2 [2,3,5,4,1] => 2 [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] => 2 [2,4,5,3,1] => 3 [2,5,1,3,4] => 2 [2,5,1,4,3] => 4 [2,5,3,1,4] => 3 [2,5,3,4,1] => 3 [2,5,4,1,3] => 4 [2,5,4,3,1] => 6 [3,1,2,4,5] => 1 [3,1,2,5,4] => 2 [3,1,4,2,5] => 2 [3,1,4,5,2] => 2 [3,1,5,2,4] => 2 [3,1,5,4,2] => 4 [3,2,1,4,5] => 2 [3,2,1,5,4] => 4 [3,2,4,1,5] => 2 [3,2,4,5,1] => 2 [3,2,5,1,4] => 4 [3,2,5,4,1] => 4 [3,4,1,2,5] => 2 [3,4,1,5,2] => 2 [3,4,2,1,5] => 3 [3,4,2,5,1] => 3 [3,4,5,1,2] => 2 [3,4,5,2,1] => 4 [3,5,1,2,4] => 2 [3,5,1,4,2] => 4 [3,5,2,1,4] => 4 [3,5,2,4,1] => 5 [3,5,4,1,2] => 4 [3,5,4,2,1] => 8 [4,1,2,3,5] => 1 [4,1,2,5,3] => 2 [4,1,3,2,5] => 2 [4,1,3,5,2] => 3 [4,1,5,2,3] => 2 [4,1,5,3,2] => 4 [4,2,1,3,5] => 2 [4,2,1,5,3] => 4 [4,2,3,1,5] => 3 [4,2,3,5,1] => 3 [4,2,5,1,3] => 4 [4,2,5,3,1] => 5 [4,3,1,2,5] => 3 [4,3,1,5,2] => 4 [4,3,2,1,5] => 6 [4,3,2,5,1] => 6 [4,3,5,1,2] => 4 [4,3,5,2,1] => 8 [4,5,1,2,3] => 2 [4,5,1,3,2] => 4 [4,5,2,1,3] => 4 [4,5,2,3,1] => 6 [4,5,3,1,2] => 6 [4,5,3,2,1] => 11 [5,1,2,3,4] => 1 [5,1,2,4,3] => 2 [5,1,3,2,4] => 2 [5,1,3,4,2] => 3 [5,1,4,2,3] => 3 [5,1,4,3,2] => 6 [5,2,1,3,4] => 2 [5,2,1,4,3] => 4 [5,2,3,1,4] => 3 [5,2,3,4,1] => 4 [5,2,4,1,3] => 5 [5,2,4,3,1] => 8 [5,3,1,2,4] => 3 [5,3,1,4,2] => 5 [5,3,2,1,4] => 6 [5,3,2,4,1] => 8 [5,3,4,1,2] => 6 [5,3,4,2,1] => 11 [5,4,1,2,3] => 4 [5,4,1,3,2] => 8 [5,4,2,1,3] => 8 [5,4,2,3,1] => 11 [5,4,3,1,2] => 11 [5,4,3,2,1] => 22 [1,2,3,4,5,6] => 1 [1,2,3,4,6,5] => 1 [1,2,3,5,4,6] => 1 [1,2,3,5,6,4] => 1 [1,2,3,6,4,5] => 1 [1,2,3,6,5,4] => 2 [1,2,4,3,5,6] => 1 [1,2,4,3,6,5] => 2 [1,2,4,5,3,6] => 1 [1,2,4,5,6,3] => 1 [1,2,4,6,3,5] => 2 [1,2,4,6,5,3] => 2 [1,2,5,3,4,6] => 1 [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] => 2 [1,2,5,6,4,3] => 3 [1,2,6,3,4,5] => 1 [1,2,6,3,5,4] => 2 [1,2,6,4,3,5] => 2 [1,2,6,4,5,3] => 3 [1,2,6,5,3,4] => 3 [1,2,6,5,4,3] => 6 [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] => 2 [1,3,2,6,4,5] => 2 [1,3,2,6,5,4] => 4 [1,3,4,2,5,6] => 1 [1,3,4,2,6,5] => 2 [1,3,4,5,2,6] => 1 [1,3,4,5,6,2] => 1 [1,3,4,6,2,5] => 2 [1,3,4,6,5,2] => 2 [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] => 2 [1,3,5,6,4,2] => 3 [1,3,6,2,4,5] => 2 [1,3,6,2,5,4] => 4 [1,3,6,4,2,5] => 3 [1,3,6,4,5,2] => 3 [1,3,6,5,2,4] => 4 [1,3,6,5,4,2] => 6 [1,4,2,3,5,6] => 1 [1,4,2,3,6,5] => 2 [1,4,2,5,3,6] => 2 [1,4,2,5,6,3] => 2 [1,4,2,6,3,5] => 2 [1,4,2,6,5,3] => 4 [1,4,3,2,5,6] => 2 [1,4,3,2,6,5] => 4 [1,4,3,5,2,6] => 2 [1,4,3,5,6,2] => 2 [1,4,3,6,2,5] => 4 [1,4,3,6,5,2] => 4 [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] => 3 [1,4,5,6,2,3] => 2 [1,4,5,6,3,2] => 4 [1,4,6,2,3,5] => 2 [1,4,6,2,5,3] => 4 [1,4,6,3,2,5] => 4 [1,4,6,3,5,2] => 5 [1,4,6,5,2,3] => 4 [1,4,6,5,3,2] => 8 [1,5,2,3,4,6] => 1 [1,5,2,3,6,4] => 2 [1,5,2,4,3,6] => 2 [1,5,2,4,6,3] => 3 [1,5,2,6,3,4] => 2 [1,5,2,6,4,3] => 4 [1,5,3,2,4,6] => 2 [1,5,3,2,6,4] => 4 [1,5,3,4,2,6] => 3 [1,5,3,4,6,2] => 3 [1,5,3,6,2,4] => 4 [1,5,3,6,4,2] => 5 [1,5,4,2,3,6] => 3 [1,5,4,2,6,3] => 4 [1,5,4,3,2,6] => 6 [1,5,4,3,6,2] => 6 [1,5,4,6,2,3] => 4 [1,5,4,6,3,2] => 8 [1,5,6,2,3,4] => 2 [1,5,6,2,4,3] => 4 [1,5,6,3,2,4] => 4 [1,5,6,3,4,2] => 6 [1,5,6,4,2,3] => 6 [1,5,6,4,3,2] => 11 [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] => 3 [1,6,2,5,3,4] => 3 [1,6,2,5,4,3] => 6 [1,6,3,2,4,5] => 2 [1,6,3,2,5,4] => 4 [1,6,3,4,2,5] => 3 [1,6,3,4,5,2] => 4 [1,6,3,5,2,4] => 5 [1,6,3,5,4,2] => 8 [1,6,4,2,3,5] => 3 [1,6,4,2,5,3] => 5 [1,6,4,3,2,5] => 6 [1,6,4,3,5,2] => 8 [1,6,4,5,2,3] => 6 [1,6,4,5,3,2] => 11 [1,6,5,2,3,4] => 4 [1,6,5,2,4,3] => 8 [1,6,5,3,2,4] => 8 [1,6,5,3,4,2] => 11 [1,6,5,4,2,3] => 11 [1,6,5,4,3,2] => 22 [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] => 2 [2,1,3,6,4,5] => 2 [2,1,3,6,5,4] => 4 [2,1,4,3,5,6] => 2 [2,1,4,3,6,5] => 3 [2,1,4,5,3,6] => 2 [2,1,4,5,6,3] => 2 [2,1,4,6,3,5] => 3 [2,1,4,6,5,3] => 4 [2,1,5,3,4,6] => 2 [2,1,5,3,6,4] => 3 [2,1,5,4,3,6] => 4 [2,1,5,4,6,3] => 4 [2,1,5,6,3,4] => 3 [2,1,5,6,4,3] => 6 [2,1,6,3,4,5] => 2 [2,1,6,3,5,4] => 4 [2,1,6,4,3,5] => 4 [2,1,6,4,5,3] => 6 [2,1,6,5,3,4] => 6 [2,1,6,5,4,3] => 11 [2,3,1,4,5,6] => 1 [2,3,1,4,6,5] => 2 [2,3,1,5,4,6] => 2 [2,3,1,5,6,4] => 3 [2,3,1,6,4,5] => 3 [2,3,1,6,5,4] => 5 [2,3,4,1,5,6] => 1 [2,3,4,1,6,5] => 2 [2,3,4,5,1,6] => 1 [2,3,4,5,6,1] => 1 [2,3,4,6,1,5] => 2 [2,3,4,6,5,1] => 2 [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] => 2 [2,3,5,6,1,4] => 3 [2,3,5,6,4,1] => 3 [2,3,6,1,4,5] => 3 [2,3,6,1,5,4] => 5 [2,3,6,4,1,5] => 3 [2,3,6,4,5,1] => 3 [2,3,6,5,1,4] => 5 [2,3,6,5,4,1] => 6 [2,4,1,3,5,6] => 2 [2,4,1,3,6,5] => 3 [2,4,1,5,3,6] => 2 [2,4,1,5,6,3] => 3 [2,4,1,6,3,5] => 3 [2,4,1,6,5,3] => 5 [2,4,3,1,5,6] => 2 [2,4,3,1,6,5] => 4 [2,4,3,5,1,6] => 2 [2,4,3,5,6,1] => 2 [2,4,3,6,1,5] => 4 [2,4,3,6,5,1] => 4 [2,4,5,1,3,6] => 2 [2,4,5,1,6,3] => 3 [2,4,5,3,1,6] => 3 [2,4,5,3,6,1] => 3 [2,4,5,6,1,3] => 3 [2,4,5,6,3,1] => 4 [2,4,6,1,3,5] => 3 [2,4,6,1,5,3] => 5 [2,4,6,3,1,5] => 5 [2,4,6,3,5,1] => 5 [2,4,6,5,1,3] => 5 [2,4,6,5,3,1] => 8 [2,5,1,3,4,6] => 2 [2,5,1,3,6,4] => 3 [2,5,1,4,3,6] => 4 [2,5,1,4,6,3] => 4 [2,5,1,6,3,4] => 3 [2,5,1,6,4,3] => 6 [2,5,3,1,4,6] => 3 [2,5,3,1,6,4] => 4 [2,5,3,4,1,6] => 3 [2,5,3,4,6,1] => 3 [2,5,3,6,1,4] => 4 [2,5,3,6,4,1] => 5 [2,5,4,1,3,6] => 4 [2,5,4,1,6,3] => 5 [2,5,4,3,1,6] => 6 [2,5,4,3,6,1] => 6 [2,5,4,6,1,3] => 5 [2,5,4,6,3,1] => 8 [2,5,6,1,3,4] => 3 [2,5,6,1,4,3] => 6 [2,5,6,3,1,4] => 5 [2,5,6,3,4,1] => 6 [2,5,6,4,1,3] => 7 [2,5,6,4,3,1] => 12 [2,6,1,3,4,5] => 2 [2,6,1,3,5,4] => 4 [2,6,1,4,3,5] => 4 [2,6,1,4,5,3] => 6 [2,6,1,5,3,4] => 6 [2,6,1,5,4,3] => 11 [2,6,3,1,4,5] => 3 [2,6,3,1,5,4] => 6 [2,6,3,4,1,5] => 4 [2,6,3,4,5,1] => 4 [2,6,3,5,1,4] => 6 [2,6,3,5,4,1] => 8 [2,6,4,1,3,5] => 5 [2,6,4,1,5,3] => 7 [2,6,4,3,1,5] => 7 [2,6,4,3,5,1] => 8 [2,6,4,5,1,3] => 7 [2,6,4,5,3,1] => 12 [2,6,5,1,3,4] => 6 [2,6,5,1,4,3] => 12 [2,6,5,3,1,4] => 9 [2,6,5,3,4,1] => 12 [2,6,5,4,1,3] => 14 [2,6,5,4,3,1] => 23 [3,1,2,4,5,6] => 1 [3,1,2,4,6,5] => 2 [3,1,2,5,4,6] => 2 [3,1,2,5,6,4] => 3 [3,1,2,6,4,5] => 3 [3,1,2,6,5,4] => 5 [3,1,4,2,5,6] => 2 [3,1,4,2,6,5] => 3 [3,1,4,5,2,6] => 2 [3,1,4,5,6,2] => 2 [3,1,4,6,2,5] => 3 [3,1,4,6,5,2] => 4 [3,1,5,2,4,6] => 2 [3,1,5,2,6,4] => 3 [3,1,5,4,2,6] => 4 [3,1,5,4,6,2] => 4 [3,1,5,6,2,4] => 3 [3,1,5,6,4,2] => 6 [3,1,6,2,4,5] => 3 [3,1,6,2,5,4] => 5 [3,1,6,4,2,5] => 4 [3,1,6,4,5,2] => 6 [3,1,6,5,2,4] => 6 [3,1,6,5,4,2] => 11 [3,2,1,4,5,6] => 2 [3,2,1,4,6,5] => 4 [3,2,1,5,4,6] => 4 [3,2,1,5,6,4] => 5 [3,2,1,6,4,5] => 5 [3,2,1,6,5,4] => 10 [3,2,4,1,5,6] => 2 [3,2,4,1,6,5] => 4 [3,2,4,5,1,6] => 2 [3,2,4,5,6,1] => 2 [3,2,4,6,1,5] => 4 [3,2,4,6,5,1] => 4 [3,2,5,1,4,6] => 4 [3,2,5,1,6,4] => 5 [3,2,5,4,1,6] => 4 [3,2,5,4,6,1] => 4 [3,2,5,6,1,4] => 5 [3,2,5,6,4,1] => 6 [3,2,6,1,4,5] => 5 [3,2,6,1,5,4] => 10 [3,2,6,4,1,5] => 6 [3,2,6,4,5,1] => 6 [3,2,6,5,1,4] => 10 [3,2,6,5,4,1] => 12 [3,4,1,2,5,6] => 2 [3,4,1,2,6,5] => 3 [3,4,1,5,2,6] => 2 [3,4,1,5,6,2] => 3 [3,4,1,6,2,5] => 3 [3,4,1,6,5,2] => 5 [3,4,2,1,5,6] => 3 [3,4,2,1,6,5] => 6 [3,4,2,5,1,6] => 3 [3,4,2,5,6,1] => 3 [3,4,2,6,1,5] => 6 [3,4,2,6,5,1] => 6 [3,4,5,1,2,6] => 2 [3,4,5,1,6,2] => 3 [3,4,5,2,1,6] => 4 [3,4,5,2,6,1] => 4 [3,4,5,6,1,2] => 3 [3,4,5,6,2,1] => 5 [3,4,6,1,2,5] => 3 [3,4,6,1,5,2] => 5 [3,4,6,2,1,5] => 6 [3,4,6,2,5,1] => 7 [3,4,6,5,1,2] => 5 [3,4,6,5,2,1] => 10 [3,5,1,2,4,6] => 2 [3,5,1,2,6,4] => 3 [3,5,1,4,2,6] => 4 [3,5,1,4,6,2] => 4 [3,5,1,6,2,4] => 3 [3,5,1,6,4,2] => 6 [3,5,2,1,4,6] => 4 [3,5,2,1,6,4] => 6 [3,5,2,4,1,6] => 5 [3,5,2,4,6,1] => 5 [3,5,2,6,1,4] => 6 [3,5,2,6,4,1] => 8 [3,5,4,1,2,6] => 4 [3,5,4,1,6,2] => 5 [3,5,4,2,1,6] => 8 [3,5,4,2,6,1] => 8 [3,5,4,6,1,2] => 5 [3,5,4,6,2,1] => 10 [3,5,6,1,2,4] => 3 [3,5,6,1,4,2] => 6 [3,5,6,2,1,4] => 6 [3,5,6,2,4,1] => 9 [3,5,6,4,1,2] => 7 [3,5,6,4,2,1] => 14 [3,6,1,2,4,5] => 3 [3,6,1,2,5,4] => 5 [3,6,1,4,2,5] => 4 [3,6,1,4,5,2] => 6 [3,6,1,5,2,4] => 6 [3,6,1,5,4,2] => 11 [3,6,2,1,4,5] => 5 [3,6,2,1,5,4] => 10 [3,6,2,4,1,5] => 6 [3,6,2,4,5,1] => 7 [3,6,2,5,1,4] => 10 [3,6,2,5,4,1] => 14 [3,6,4,1,2,5] => 5 [3,6,4,1,5,2] => 7 [3,6,4,2,1,5] => 9 [3,6,4,2,5,1] => 11 [3,6,4,5,1,2] => 7 [3,6,4,5,2,1] => 14 [3,6,5,1,2,4] => 6 [3,6,5,1,4,2] => 12 [3,6,5,2,1,4] => 12 [3,6,5,2,4,1] => 17 [3,6,5,4,1,2] => 14 [3,6,5,4,2,1] => 28 [4,1,2,3,5,6] => 1 [4,1,2,3,6,5] => 2 [4,1,2,5,3,6] => 2 [4,1,2,5,6,3] => 3 [4,1,2,6,3,5] => 3 [4,1,2,6,5,3] => 5 [4,1,3,2,5,6] => 2 [4,1,3,2,6,5] => 4 [4,1,3,5,2,6] => 3 [4,1,3,5,6,2] => 3 [4,1,3,6,2,5] => 4 [4,1,3,6,5,2] => 6 [4,1,5,2,3,6] => 2 [4,1,5,2,6,3] => 3 [4,1,5,3,2,6] => 4 [4,1,5,3,6,2] => 5 [4,1,5,6,2,3] => 3 [4,1,5,6,3,2] => 6 [4,1,6,2,3,5] => 3 [4,1,6,2,5,3] => 5 [4,1,6,3,2,5] => 5 [4,1,6,3,5,2] => 7 [4,1,6,5,2,3] => 6 [4,1,6,5,3,2] => 12 [4,2,1,3,5,6] => 2 [4,2,1,3,6,5] => 4 [4,2,1,5,3,6] => 4 [4,2,1,5,6,3] => 5 [4,2,1,6,3,5] => 5 [4,2,1,6,5,3] => 10 [4,2,3,1,5,6] => 3 [4,2,3,1,6,5] => 6 [4,2,3,5,1,6] => 3 [4,2,3,5,6,1] => 3 [4,2,3,6,1,5] => 6 [4,2,3,6,5,1] => 6 [4,2,5,1,3,6] => 4 [4,2,5,1,6,3] => 5 [4,2,5,3,1,6] => 5 [4,2,5,3,6,1] => 5 [4,2,5,6,1,3] => 5 [4,2,5,6,3,1] => 7 [4,2,6,1,3,5] => 5 [4,2,6,1,5,3] => 10 [4,2,6,3,1,5] => 7 [4,2,6,3,5,1] => 8 [4,2,6,5,1,3] => 10 [4,2,6,5,3,1] => 14 [4,3,1,2,5,6] => 3 [4,3,1,2,6,5] => 6 [4,3,1,5,2,6] => 4 [4,3,1,5,6,2] => 5 [4,3,1,6,2,5] => 6 [4,3,1,6,5,2] => 10 [4,3,2,1,5,6] => 6 [4,3,2,1,6,5] => 11 [4,3,2,5,1,6] => 6 [4,3,2,5,6,1] => 6 [4,3,2,6,1,5] => 11 [4,3,2,6,5,1] => 12 [4,3,5,1,2,6] => 4 [4,3,5,1,6,2] => 5 [4,3,5,2,1,6] => 8 [4,3,5,2,6,1] => 8 [4,3,5,6,1,2] => 5 [4,3,5,6,2,1] => 10 [4,3,6,1,2,5] => 6 [4,3,6,1,5,2] => 10 [4,3,6,2,1,5] => 12 [4,3,6,2,5,1] => 14 [4,3,6,5,1,2] => 10 [4,3,6,5,2,1] => 19 [4,5,1,2,3,6] => 2 [4,5,1,2,6,3] => 3 [4,5,1,3,2,6] => 4 [4,5,1,3,6,2] => 5 [4,5,1,6,2,3] => 3 [4,5,1,6,3,2] => 6 [4,5,2,1,3,6] => 4 [4,5,2,1,6,3] => 6 [4,5,2,3,1,6] => 6 [4,5,2,3,6,1] => 6 [4,5,2,6,1,3] => 6 [4,5,2,6,3,1] => 9 [4,5,3,1,2,6] => 6 [4,5,3,1,6,2] => 7 [4,5,3,2,1,6] => 11 [4,5,3,2,6,1] => 12 [4,5,3,6,1,2] => 7 [4,5,3,6,2,1] => 14 [4,5,6,1,2,3] => 3 [4,5,6,1,3,2] => 6 [4,5,6,2,1,3] => 6 [4,5,6,2,3,1] => 9 [4,5,6,3,1,2] => 9 [4,5,6,3,2,1] => 18 [4,6,1,2,3,5] => 3 [4,6,1,2,5,3] => 5 [4,6,1,3,2,5] => 5 [4,6,1,3,5,2] => 7 [4,6,1,5,2,3] => 6 [4,6,1,5,3,2] => 12 [4,6,2,1,3,5] => 5 [4,6,2,1,5,3] => 10 [4,6,2,3,1,5] => 7 [4,6,2,3,5,1] => 9 [4,6,2,5,1,3] => 10 [4,6,2,5,3,1] => 15 [4,6,3,1,2,5] => 7 [4,6,3,1,5,2] => 12 [4,6,3,2,1,5] => 14 [4,6,3,2,5,1] => 17 [4,6,3,5,1,2] => 12 [4,6,3,5,2,1] => 23 [4,6,5,1,2,3] => 6 [4,6,5,1,3,2] => 12 [4,6,5,2,1,3] => 12 [4,6,5,2,3,1] => 18 [4,6,5,3,1,2] => 18 [4,6,5,3,2,1] => 35 [5,1,2,3,4,6] => 1 [5,1,2,3,6,4] => 2 [5,1,2,4,3,6] => 2 [5,1,2,4,6,3] => 3 [5,1,2,6,3,4] => 3 [5,1,2,6,4,3] => 5 [5,1,3,2,4,6] => 2 [5,1,3,2,6,4] => 4 [5,1,3,4,2,6] => 3 [5,1,3,4,6,2] => 4 [5,1,3,6,2,4] => 4 [5,1,3,6,4,2] => 6 [5,1,4,2,3,6] => 3 [5,1,4,2,6,3] => 5 [5,1,4,3,2,6] => 6 [5,1,4,3,6,2] => 7 [5,1,4,6,2,3] => 5 [5,1,4,6,3,2] => 9 [5,1,6,2,3,4] => 3 [5,1,6,2,4,3] => 5 [5,1,6,3,2,4] => 5 [5,1,6,3,4,2] => 7 [5,1,6,4,2,3] => 7 [5,1,6,4,3,2] => 14 [5,2,1,3,4,6] => 2 [5,2,1,3,6,4] => 4 [5,2,1,4,3,6] => 4 [5,2,1,4,6,3] => 6 [5,2,1,6,3,4] => 5 [5,2,1,6,4,3] => 10 [5,2,3,1,4,6] => 3 [5,2,3,1,6,4] => 6 [5,2,3,4,1,6] => 4 [5,2,3,4,6,1] => 4 [5,2,3,6,1,4] => 6 [5,2,3,6,4,1] => 7 [5,2,4,1,3,6] => 5 [5,2,4,1,6,3] => 7 [5,2,4,3,1,6] => 8 [5,2,4,3,6,1] => 8 [5,2,4,6,1,3] => 7 [5,2,4,6,3,1] => 11 [5,2,6,1,3,4] => 5 [5,2,6,1,4,3] => 10 [5,2,6,3,1,4] => 7 [5,2,6,3,4,1] => 9 [5,2,6,4,1,3] => 12 [5,2,6,4,3,1] => 17 [5,3,1,2,4,6] => 3 [5,3,1,2,6,4] => 6 [5,3,1,4,2,6] => 5 [5,3,1,4,6,2] => 6 [5,3,1,6,2,4] => 6 [5,3,1,6,4,2] => 10 [5,3,2,1,4,6] => 6 [5,3,2,1,6,4] => 11 [5,3,2,4,1,6] => 8 [5,3,2,4,6,1] => 8 [5,3,2,6,1,4] => 11 [5,3,2,6,4,1] => 14 [5,3,4,1,2,6] => 6 [5,3,4,1,6,2] => 7 [5,3,4,2,1,6] => 11 [5,3,4,2,6,1] => 12 [5,3,4,6,1,2] => 7 [5,3,4,6,2,1] => 14 [5,3,6,1,2,4] => 6 [5,3,6,1,4,2] => 10 [5,3,6,2,1,4] => 12 [5,3,6,2,4,1] => 15 [5,3,6,4,1,2] => 12 [5,3,6,4,2,1] => 23 [5,4,1,2,3,6] => 4 [5,4,1,2,6,3] => 6 [5,4,1,3,2,6] => 8 [5,4,1,3,6,2] => 9 [5,4,1,6,2,3] => 6 [5,4,1,6,3,2] => 12 [5,4,2,1,3,6] => 8 [5,4,2,1,6,3] => 12 [5,4,2,3,1,6] => 11 [5,4,2,3,6,1] => 12 [5,4,2,6,1,3] => 12 [5,4,2,6,3,1] => 17 [5,4,3,1,2,6] => 11 [5,4,3,1,6,2] => 14 [5,4,3,2,1,6] => 22 [5,4,3,2,6,1] => 23 [5,4,3,6,1,2] => 14 [5,4,3,6,2,1] => 28 [5,4,6,1,2,3] => 6 [5,4,6,1,3,2] => 12 [5,4,6,2,1,3] => 12 [5,4,6,2,3,1] => 18 [5,4,6,3,1,2] => 18 [5,4,6,3,2,1] => 35 [5,6,1,2,3,4] => 3 [5,6,1,2,4,3] => 5 [5,6,1,3,2,4] => 5 [5,6,1,3,4,2] => 7 [5,6,1,4,2,3] => 7 [5,6,1,4,3,2] => 14 [5,6,2,1,3,4] => 5 [5,6,2,1,4,3] => 10 [5,6,2,3,1,4] => 7 [5,6,2,3,4,1] => 9 [5,6,2,4,1,3] => 12 [5,6,2,4,3,1] => 18 [5,6,3,1,2,4] => 7 [5,6,3,1,4,2] => 12 [5,6,3,2,1,4] => 14 [5,6,3,2,4,1] => 18 [5,6,3,4,1,2] => 14 [5,6,3,4,2,1] => 26 [5,6,4,1,2,3] => 9 [5,6,4,1,3,2] => 18 [5,6,4,2,1,3] => 18 [5,6,4,2,3,1] => 26 [5,6,4,3,1,2] => 26 [5,6,4,3,2,1] => 52 [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] => 3 [6,1,2,5,3,4] => 3 [6,1,2,5,4,3] => 6 [6,1,3,2,4,5] => 2 [6,1,3,2,5,4] => 4 [6,1,3,4,2,5] => 3 [6,1,3,4,5,2] => 4 [6,1,3,5,2,4] => 5 [6,1,3,5,4,2] => 8 [6,1,4,2,3,5] => 3 [6,1,4,2,5,3] => 5 [6,1,4,3,2,5] => 6 [6,1,4,3,5,2] => 8 [6,1,4,5,2,3] => 6 [6,1,4,5,3,2] => 12 [6,1,5,2,3,4] => 4 [6,1,5,2,4,3] => 8 [6,1,5,3,2,4] => 8 [6,1,5,3,4,2] => 12 [6,1,5,4,2,3] => 12 [6,1,5,4,3,2] => 23 [6,2,1,3,4,5] => 2 [6,2,1,3,5,4] => 4 [6,2,1,4,3,5] => 4 [6,2,1,4,5,3] => 6 [6,2,1,5,3,4] => 6 [6,2,1,5,4,3] => 12 [6,2,3,1,4,5] => 3 [6,2,3,1,5,4] => 6 [6,2,3,4,1,5] => 4 [6,2,3,4,5,1] => 5 [6,2,3,5,1,4] => 7 [6,2,3,5,4,1] => 10 [6,2,4,1,3,5] => 5 [6,2,4,1,5,3] => 8 [6,2,4,3,1,5] => 8 [6,2,4,3,5,1] => 10 [6,2,4,5,1,3] => 9 [6,2,4,5,3,1] => 14 [6,2,5,1,3,4] => 7 [6,2,5,1,4,3] => 14 [6,2,5,3,1,4] => 11 [6,2,5,3,4,1] => 14 [6,2,5,4,1,3] => 17 [6,2,5,4,3,1] => 28 [6,3,1,2,4,5] => 3 [6,3,1,2,5,4] => 6 [6,3,1,4,2,5] => 5 [6,3,1,4,5,2] => 7 [6,3,1,5,2,4] => 8 [6,3,1,5,4,2] => 14 [6,3,2,1,4,5] => 6 [6,3,2,1,5,4] => 12 [6,3,2,4,1,5] => 8 [6,3,2,4,5,1] => 10 [6,3,2,5,1,4] => 14 [6,3,2,5,4,1] => 19 [6,3,4,1,2,5] => 6 [6,3,4,1,5,2] => 9 [6,3,4,2,1,5] => 12 [6,3,4,2,5,1] => 14 [6,3,4,5,1,2] => 9 [6,3,4,5,2,1] => 18 [6,3,5,1,2,4] => 9 [6,3,5,1,4,2] => 15 [6,3,5,2,1,4] => 17 [6,3,5,2,4,1] => 23 [6,3,5,4,1,2] => 18 [6,3,5,4,2,1] => 35 [6,4,1,2,3,5] => 4 [6,4,1,2,5,3] => 7 [6,4,1,3,2,5] => 8 [6,4,1,3,5,2] => 11 [6,4,1,5,2,3] => 9 [6,4,1,5,3,2] => 17 [6,4,2,1,3,5] => 8 [6,4,2,1,5,3] => 14 [6,4,2,3,1,5] => 12 [6,4,2,3,5,1] => 14 [6,4,2,5,1,3] => 15 [6,4,2,5,3,1] => 23 [6,4,3,1,2,5] => 12 [6,4,3,1,5,2] => 17 [6,4,3,2,1,5] => 23 [6,4,3,2,5,1] => 28 [6,4,3,5,1,2] => 18 [6,4,3,5,2,1] => 35 [6,4,5,1,2,3] => 9 [6,4,5,1,3,2] => 18 [6,4,5,2,1,3] => 18 [6,4,5,2,3,1] => 26 [6,4,5,3,1,2] => 26 [6,4,5,3,2,1] => 52 [6,5,1,2,3,4] => 5 [6,5,1,2,4,3] => 10 [6,5,1,3,2,4] => 10 [6,5,1,3,4,2] => 14 [6,5,1,4,2,3] => 14 [6,5,1,4,3,2] => 28 [6,5,2,1,3,4] => 10 [6,5,2,1,4,3] => 19 [6,5,2,3,1,4] => 14 [6,5,2,3,4,1] => 18 [6,5,2,4,1,3] => 23 [6,5,2,4,3,1] => 35 [6,5,3,1,2,4] => 14 [6,5,3,1,4,2] => 23 [6,5,3,2,1,4] => 28 [6,5,3,2,4,1] => 35 [6,5,3,4,1,2] => 26 [6,5,3,4,2,1] => 52 [6,5,4,1,2,3] => 18 [6,5,4,1,3,2] => 35 [6,5,4,2,1,3] => 35 [6,5,4,2,3,1] => 52 [6,5,4,3,1,2] => 52 [6,5,4,3,2,1] => 101 ----------------------------------------------------------------------------- Created: Jun 26, 2018 at 12:54 by Susanna Fishel ----------------------------------------------------------------------------- Last Updated: Jun 26, 2018 at 13:13 by Christian Stump