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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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* This information is distributed in the hope that it will be useful, *
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-----------------------------------------------------------------------------
Statistic identifier: St000156
-----------------------------------------------------------------------------
Collection: Permutations
-----------------------------------------------------------------------------
Description: The Denert index of a permutation.
It is defined as
$$
\begin{align*}
den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\
&+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\
&+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \}
\end{align*}
$$
where $n$ is the size of $\sigma$. It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $exc$ is the number of weak exceedences, see [[St000155]].
-----------------------------------------------------------------------------
References: [1] Denert, M. The genus zeta function of hereditary orders in central simple algebras over global fields [[MathSciNet:0993928]]
[2] Foata, D., Zeilberger, D. Denert's permutation statistic is indeed Euler-Mahonian [[MathSciNet:1061147]]
[3] Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_i=0..n-1 (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). [[OEIS:A008302]]
-----------------------------------------------------------------------------
Code:
import itertools
def statistic(pi):
return sum(1 for l, k in itertools.combinations(range(pi.size()), 2)
if (pi[k] < pi[l] <= k+1)
or (pi[l] <= k+1 < pi[k])
or (k+1 < pi[k] < pi[l]))
-----------------------------------------------------------------------------
Statistic values:
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 3
[3,1,2] => 1
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 3
[1,3,2,4] => 2
[1,3,4,2] => 5
[1,4,2,3] => 2
[1,4,3,2] => 3
[2,1,3,4] => 1
[2,1,4,3] => 4
[2,3,1,4] => 3
[2,3,4,1] => 6
[2,4,1,3] => 3
[2,4,3,1] => 4
[3,1,2,4] => 1
[3,1,4,2] => 4
[3,2,1,4] => 2
[3,2,4,1] => 5
[3,4,1,2] => 3
[3,4,2,1] => 4
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 3
[4,3,1,2] => 4
[4,3,2,1] => 5
[1,2,3,4,5] => 0
[1,2,3,5,4] => 4
[1,2,4,3,5] => 3
[1,2,4,5,3] => 7
[1,2,5,3,4] => 3
[1,2,5,4,3] => 4
[1,3,2,4,5] => 2
[1,3,2,5,4] => 6
[1,3,4,2,5] => 5
[1,3,4,5,2] => 9
[1,3,5,2,4] => 5
[1,3,5,4,2] => 6
[1,4,2,3,5] => 2
[1,4,2,5,3] => 6
[1,4,3,2,5] => 3
[1,4,3,5,2] => 7
[1,4,5,2,3] => 5
[1,4,5,3,2] => 6
[1,5,2,3,4] => 2
[1,5,2,4,3] => 3
[1,5,3,2,4] => 3
[1,5,3,4,2] => 4
[1,5,4,2,3] => 6
[1,5,4,3,2] => 7
[2,1,3,4,5] => 1
[2,1,3,5,4] => 5
[2,1,4,3,5] => 4
[2,1,4,5,3] => 8
[2,1,5,3,4] => 4
[2,1,5,4,3] => 5
[2,3,1,4,5] => 3
[2,3,1,5,4] => 7
[2,3,4,1,5] => 6
[2,3,4,5,1] => 10
[2,3,5,1,4] => 6
[2,3,5,4,1] => 7
[2,4,1,3,5] => 3
[2,4,1,5,3] => 7
[2,4,3,1,5] => 4
[2,4,3,5,1] => 8
[2,4,5,1,3] => 6
[2,4,5,3,1] => 7
[2,5,1,3,4] => 3
[2,5,1,4,3] => 4
[2,5,3,1,4] => 4
[2,5,3,4,1] => 5
[2,5,4,1,3] => 7
[2,5,4,3,1] => 8
[3,1,2,4,5] => 1
[3,1,2,5,4] => 5
[3,1,4,2,5] => 4
[3,1,4,5,2] => 8
[3,1,5,2,4] => 4
[3,1,5,4,2] => 5
[3,2,1,4,5] => 2
[3,2,1,5,4] => 6
[3,2,4,1,5] => 5
[3,2,4,5,1] => 9
[3,2,5,1,4] => 5
[3,2,5,4,1] => 6
[3,4,1,2,5] => 3
[3,4,1,5,2] => 7
[3,4,2,1,5] => 4
[3,4,2,5,1] => 8
[3,4,5,1,2] => 6
[3,4,5,2,1] => 7
[3,5,1,2,4] => 3
[3,5,1,4,2] => 4
[3,5,2,1,4] => 4
[3,5,2,4,1] => 5
[3,5,4,1,2] => 7
[3,5,4,2,1] => 8
[4,1,2,3,5] => 1
[4,1,2,5,3] => 5
[4,1,3,2,5] => 2
[4,1,3,5,2] => 6
[4,1,5,2,3] => 4
[4,1,5,3,2] => 5
[4,2,1,3,5] => 2
[4,2,1,5,3] => 6
[4,2,3,1,5] => 3
[4,2,3,5,1] => 7
[4,2,5,1,3] => 5
[4,2,5,3,1] => 6
[4,3,1,2,5] => 4
[4,3,1,5,2] => 8
[4,3,2,1,5] => 5
[4,3,2,5,1] => 9
[4,3,5,1,2] => 7
[4,3,5,2,1] => 8
[4,5,1,2,3] => 3
[4,5,1,3,2] => 4
[4,5,2,1,3] => 4
[4,5,2,3,1] => 5
[4,5,3,1,2] => 5
[4,5,3,2,1] => 6
[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] => 5
[5,1,4,3,2] => 6
[5,2,1,3,4] => 2
[5,2,1,4,3] => 3
[5,2,3,1,4] => 3
[5,2,3,4,1] => 4
[5,2,4,1,3] => 6
[5,2,4,3,1] => 7
[5,3,1,2,4] => 4
[5,3,1,4,2] => 5
[5,3,2,1,4] => 5
[5,3,2,4,1] => 6
[5,3,4,1,2] => 8
[5,3,4,2,1] => 9
[5,4,1,2,3] => 4
[5,4,1,3,2] => 5
[5,4,2,1,3] => 5
[5,4,2,3,1] => 6
[5,4,3,1,2] => 6
[5,4,3,2,1] => 7
[1,2,3,4,5,6] => 0
[1,2,3,4,6,5] => 5
[1,2,3,5,4,6] => 4
[1,2,3,5,6,4] => 9
[1,2,3,6,4,5] => 4
[1,2,3,6,5,4] => 5
[1,2,4,3,5,6] => 3
[1,2,4,3,6,5] => 8
[1,2,4,5,3,6] => 7
[1,2,4,5,6,3] => 12
[1,2,4,6,3,5] => 7
[1,2,4,6,5,3] => 8
[1,2,5,3,4,6] => 3
[1,2,5,3,6,4] => 8
[1,2,5,4,3,6] => 4
[1,2,5,4,6,3] => 9
[1,2,5,6,3,4] => 7
[1,2,5,6,4,3] => 8
[1,2,6,3,4,5] => 3
[1,2,6,3,5,4] => 4
[1,2,6,4,3,5] => 4
[1,2,6,4,5,3] => 5
[1,2,6,5,3,4] => 8
[1,2,6,5,4,3] => 9
[1,3,2,4,5,6] => 2
[1,3,2,4,6,5] => 7
[1,3,2,5,4,6] => 6
[1,3,2,5,6,4] => 11
[1,3,2,6,4,5] => 6
[1,3,2,6,5,4] => 7
[1,3,4,2,5,6] => 5
[1,3,4,2,6,5] => 10
[1,3,4,5,2,6] => 9
[1,3,4,5,6,2] => 14
[1,3,4,6,2,5] => 9
[1,3,4,6,5,2] => 10
[1,3,5,2,4,6] => 5
[1,3,5,2,6,4] => 10
[1,3,5,4,2,6] => 6
[1,3,5,4,6,2] => 11
[1,3,5,6,2,4] => 9
[1,3,5,6,4,2] => 10
[1,3,6,2,4,5] => 5
[1,3,6,2,5,4] => 6
[1,3,6,4,2,5] => 6
[1,3,6,4,5,2] => 7
[1,3,6,5,2,4] => 10
[1,3,6,5,4,2] => 11
[1,4,2,3,5,6] => 2
[1,4,2,3,6,5] => 7
[1,4,2,5,3,6] => 6
[1,4,2,5,6,3] => 11
[1,4,2,6,3,5] => 6
[1,4,2,6,5,3] => 7
[1,4,3,2,5,6] => 3
[1,4,3,2,6,5] => 8
[1,4,3,5,2,6] => 7
[1,4,3,5,6,2] => 12
[1,4,3,6,2,5] => 7
[1,4,3,6,5,2] => 8
[1,4,5,2,3,6] => 5
[1,4,5,2,6,3] => 10
[1,4,5,3,2,6] => 6
[1,4,5,3,6,2] => 11
[1,4,5,6,2,3] => 9
[1,4,5,6,3,2] => 10
[1,4,6,2,3,5] => 5
[1,4,6,2,5,3] => 6
[1,4,6,3,2,5] => 6
[1,4,6,3,5,2] => 7
[1,4,6,5,2,3] => 10
[1,4,6,5,3,2] => 11
[1,5,2,3,4,6] => 2
[1,5,2,3,6,4] => 7
[1,5,2,4,3,6] => 3
[1,5,2,4,6,3] => 8
[1,5,2,6,3,4] => 6
[1,5,2,6,4,3] => 7
[1,5,3,2,4,6] => 3
[1,5,3,2,6,4] => 8
[1,5,3,4,2,6] => 4
[1,5,3,4,6,2] => 9
[1,5,3,6,2,4] => 7
[1,5,3,6,4,2] => 8
[1,5,4,2,3,6] => 6
[1,5,4,2,6,3] => 11
[1,5,4,3,2,6] => 7
[1,5,4,3,6,2] => 12
[1,5,4,6,2,3] => 10
[1,5,4,6,3,2] => 11
[1,5,6,2,3,4] => 5
[1,5,6,2,4,3] => 6
[1,5,6,3,2,4] => 6
[1,5,6,3,4,2] => 7
[1,5,6,4,2,3] => 7
[1,5,6,4,3,2] => 8
[1,6,2,3,4,5] => 2
[1,6,2,3,5,4] => 3
[1,6,2,4,3,5] => 3
[1,6,2,4,5,3] => 4
[1,6,2,5,3,4] => 7
[1,6,2,5,4,3] => 8
[1,6,3,2,4,5] => 3
[1,6,3,2,5,4] => 4
[1,6,3,4,2,5] => 4
[1,6,3,4,5,2] => 5
[1,6,3,5,2,4] => 8
[1,6,3,5,4,2] => 9
[1,6,4,2,3,5] => 6
[1,6,4,2,5,3] => 7
[1,6,4,3,2,5] => 7
[1,6,4,3,5,2] => 8
[1,6,4,5,2,3] => 11
[1,6,4,5,3,2] => 12
[1,6,5,2,3,4] => 6
[1,6,5,2,4,3] => 7
[1,6,5,3,2,4] => 7
[1,6,5,3,4,2] => 8
[1,6,5,4,2,3] => 8
[1,6,5,4,3,2] => 9
[2,1,3,4,5,6] => 1
[2,1,3,4,6,5] => 6
[2,1,3,5,4,6] => 5
[2,1,3,5,6,4] => 10
[2,1,3,6,4,5] => 5
[2,1,3,6,5,4] => 6
[2,1,4,3,5,6] => 4
[2,1,4,3,6,5] => 9
[2,1,4,5,3,6] => 8
[2,1,4,5,6,3] => 13
[2,1,4,6,3,5] => 8
[2,1,4,6,5,3] => 9
[2,1,5,3,4,6] => 4
[2,1,5,3,6,4] => 9
[2,1,5,4,3,6] => 5
[2,1,5,4,6,3] => 10
[2,1,5,6,3,4] => 8
[2,1,5,6,4,3] => 9
[2,1,6,3,4,5] => 4
[2,1,6,3,5,4] => 5
[2,1,6,4,3,5] => 5
[2,1,6,4,5,3] => 6
[2,1,6,5,3,4] => 9
[2,1,6,5,4,3] => 10
[2,3,1,4,5,6] => 3
[2,3,1,4,6,5] => 8
[2,3,1,5,4,6] => 7
[2,3,1,5,6,4] => 12
[2,3,1,6,4,5] => 7
[2,3,1,6,5,4] => 8
[2,3,4,1,5,6] => 6
[2,3,4,1,6,5] => 11
[2,3,4,5,1,6] => 10
[2,3,4,5,6,1] => 15
[2,3,4,6,1,5] => 10
[2,3,4,6,5,1] => 11
[2,3,5,1,4,6] => 6
[2,3,5,1,6,4] => 11
[2,3,5,4,1,6] => 7
[2,3,5,4,6,1] => 12
[2,3,5,6,1,4] => 10
[2,3,5,6,4,1] => 11
[2,3,6,1,4,5] => 6
[2,3,6,1,5,4] => 7
[2,3,6,4,1,5] => 7
[2,3,6,4,5,1] => 8
[2,3,6,5,1,4] => 11
[2,3,6,5,4,1] => 12
[2,4,1,3,5,6] => 3
[2,4,1,3,6,5] => 8
[2,4,1,5,3,6] => 7
[2,4,1,5,6,3] => 12
[2,4,1,6,3,5] => 7
[2,4,1,6,5,3] => 8
[2,4,3,1,5,6] => 4
[2,4,3,1,6,5] => 9
[2,4,3,5,1,6] => 8
[2,4,3,5,6,1] => 13
[2,4,3,6,1,5] => 8
[2,4,3,6,5,1] => 9
[2,4,5,1,3,6] => 6
[2,4,5,1,6,3] => 11
[2,4,5,3,1,6] => 7
[2,4,5,3,6,1] => 12
[2,4,5,6,1,3] => 10
[2,4,5,6,3,1] => 11
[2,4,6,1,3,5] => 6
[2,4,6,1,5,3] => 7
[2,4,6,3,1,5] => 7
[2,4,6,3,5,1] => 8
[2,4,6,5,1,3] => 11
[2,4,6,5,3,1] => 12
[2,5,1,3,4,6] => 3
[2,5,1,3,6,4] => 8
[2,5,1,4,3,6] => 4
[2,5,1,4,6,3] => 9
[2,5,1,6,3,4] => 7
[2,5,1,6,4,3] => 8
[2,5,3,1,4,6] => 4
[2,5,3,1,6,4] => 9
[2,5,3,4,1,6] => 5
[2,5,3,4,6,1] => 10
[2,5,3,6,1,4] => 8
[2,5,3,6,4,1] => 9
[2,5,4,1,3,6] => 7
[2,5,4,1,6,3] => 12
[2,5,4,3,1,6] => 8
[2,5,4,3,6,1] => 13
[2,5,4,6,1,3] => 11
[2,5,4,6,3,1] => 12
[2,5,6,1,3,4] => 6
[2,5,6,1,4,3] => 7
[2,5,6,3,1,4] => 7
[2,5,6,3,4,1] => 8
[2,5,6,4,1,3] => 8
[2,5,6,4,3,1] => 9
[2,6,1,3,4,5] => 3
[2,6,1,3,5,4] => 4
[2,6,1,4,3,5] => 4
[2,6,1,4,5,3] => 5
[2,6,1,5,3,4] => 8
[2,6,1,5,4,3] => 9
[2,6,3,1,4,5] => 4
[2,6,3,1,5,4] => 5
[2,6,3,4,1,5] => 5
[2,6,3,4,5,1] => 6
[2,6,3,5,1,4] => 9
[2,6,3,5,4,1] => 10
[2,6,4,1,3,5] => 7
[2,6,4,1,5,3] => 8
[2,6,4,3,1,5] => 8
[2,6,4,3,5,1] => 9
[2,6,4,5,1,3] => 12
[2,6,4,5,3,1] => 13
[2,6,5,1,3,4] => 7
[2,6,5,1,4,3] => 8
[2,6,5,3,1,4] => 8
[2,6,5,3,4,1] => 9
[2,6,5,4,1,3] => 9
[2,6,5,4,3,1] => 10
[3,1,2,4,5,6] => 1
[3,1,2,4,6,5] => 6
[3,1,2,5,4,6] => 5
[3,1,2,5,6,4] => 10
[3,1,2,6,4,5] => 5
[3,1,2,6,5,4] => 6
[3,1,4,2,5,6] => 4
[3,1,4,2,6,5] => 9
[3,1,4,5,2,6] => 8
[3,1,4,5,6,2] => 13
[3,1,4,6,2,5] => 8
[3,1,4,6,5,2] => 9
[3,1,5,2,4,6] => 4
[3,1,5,2,6,4] => 9
[3,1,5,4,2,6] => 5
[3,1,5,4,6,2] => 10
[3,1,5,6,2,4] => 8
[3,1,5,6,4,2] => 9
[3,1,6,2,4,5] => 4
[3,1,6,2,5,4] => 5
[3,1,6,4,2,5] => 5
[3,1,6,4,5,2] => 6
[3,1,6,5,2,4] => 9
[3,1,6,5,4,2] => 10
[3,2,1,4,5,6] => 2
[3,2,1,4,6,5] => 7
[3,2,1,5,4,6] => 6
[3,2,1,5,6,4] => 11
[3,2,1,6,4,5] => 6
[3,2,1,6,5,4] => 7
[3,2,4,1,5,6] => 5
[3,2,4,1,6,5] => 10
[3,2,4,5,1,6] => 9
[3,2,4,5,6,1] => 14
[3,2,4,6,1,5] => 9
[3,2,4,6,5,1] => 10
[3,2,5,1,4,6] => 5
[3,2,5,1,6,4] => 10
[3,2,5,4,1,6] => 6
[3,2,5,4,6,1] => 11
[3,2,5,6,1,4] => 9
[3,2,5,6,4,1] => 10
[3,2,6,1,4,5] => 5
[3,2,6,1,5,4] => 6
[3,2,6,4,1,5] => 6
[3,2,6,4,5,1] => 7
[3,2,6,5,1,4] => 10
[3,2,6,5,4,1] => 11
[3,4,1,2,5,6] => 3
[3,4,1,2,6,5] => 8
[3,4,1,5,2,6] => 7
[3,4,1,5,6,2] => 12
[3,4,1,6,2,5] => 7
[3,4,1,6,5,2] => 8
[3,4,2,1,5,6] => 4
[3,4,2,1,6,5] => 9
[3,4,2,5,1,6] => 8
[3,4,2,5,6,1] => 13
[3,4,2,6,1,5] => 8
[3,4,2,6,5,1] => 9
[3,4,5,1,2,6] => 6
[3,4,5,1,6,2] => 11
[3,4,5,2,1,6] => 7
[3,4,5,2,6,1] => 12
[3,4,5,6,1,2] => 10
[3,4,5,6,2,1] => 11
[3,4,6,1,2,5] => 6
[3,4,6,1,5,2] => 7
[3,4,6,2,1,5] => 7
[3,4,6,2,5,1] => 8
[3,4,6,5,1,2] => 11
[3,4,6,5,2,1] => 12
[3,5,1,2,4,6] => 3
[3,5,1,2,6,4] => 8
[3,5,1,4,2,6] => 4
[3,5,1,4,6,2] => 9
[3,5,1,6,2,4] => 7
[3,5,1,6,4,2] => 8
[3,5,2,1,4,6] => 4
[3,5,2,1,6,4] => 9
[3,5,2,4,1,6] => 5
[3,5,2,4,6,1] => 10
[3,5,2,6,1,4] => 8
[3,5,2,6,4,1] => 9
[3,5,4,1,2,6] => 7
[3,5,4,1,6,2] => 12
[3,5,4,2,1,6] => 8
[3,5,4,2,6,1] => 13
[3,5,4,6,1,2] => 11
[3,5,4,6,2,1] => 12
[3,5,6,1,2,4] => 6
[3,5,6,1,4,2] => 7
[3,5,6,2,1,4] => 7
[3,5,6,2,4,1] => 8
[3,5,6,4,1,2] => 8
[3,5,6,4,2,1] => 9
[3,6,1,2,4,5] => 3
[3,6,1,2,5,4] => 4
[3,6,1,4,2,5] => 4
[3,6,1,4,5,2] => 5
[3,6,1,5,2,4] => 8
[3,6,1,5,4,2] => 9
[3,6,2,1,4,5] => 4
[3,6,2,1,5,4] => 5
[3,6,2,4,1,5] => 5
[3,6,2,4,5,1] => 6
[3,6,2,5,1,4] => 9
[3,6,2,5,4,1] => 10
[3,6,4,1,2,5] => 7
[3,6,4,1,5,2] => 8
[3,6,4,2,1,5] => 8
[3,6,4,2,5,1] => 9
[3,6,4,5,1,2] => 12
[3,6,4,5,2,1] => 13
[3,6,5,1,2,4] => 7
[3,6,5,1,4,2] => 8
[3,6,5,2,1,4] => 8
[3,6,5,2,4,1] => 9
[3,6,5,4,1,2] => 9
[3,6,5,4,2,1] => 10
[4,1,2,3,5,6] => 1
[4,1,2,3,6,5] => 6
[4,1,2,5,3,6] => 5
[4,1,2,5,6,3] => 10
[4,1,2,6,3,5] => 5
[4,1,2,6,5,3] => 6
[4,1,3,2,5,6] => 2
[4,1,3,2,6,5] => 7
[4,1,3,5,2,6] => 6
[4,1,3,5,6,2] => 11
[4,1,3,6,2,5] => 6
[4,1,3,6,5,2] => 7
[4,1,5,2,3,6] => 4
[4,1,5,2,6,3] => 9
[4,1,5,3,2,6] => 5
[4,1,5,3,6,2] => 10
[4,1,5,6,2,3] => 8
[4,1,5,6,3,2] => 9
[4,1,6,2,3,5] => 4
[4,1,6,2,5,3] => 5
[4,1,6,3,2,5] => 5
[4,1,6,3,5,2] => 6
[4,1,6,5,2,3] => 9
[4,1,6,5,3,2] => 10
[4,2,1,3,5,6] => 2
[4,2,1,3,6,5] => 7
[4,2,1,5,3,6] => 6
[4,2,1,5,6,3] => 11
[4,2,1,6,3,5] => 6
[4,2,1,6,5,3] => 7
[4,2,3,1,5,6] => 3
[4,2,3,1,6,5] => 8
[4,2,3,5,1,6] => 7
[4,2,3,5,6,1] => 12
[4,2,3,6,1,5] => 7
[4,2,3,6,5,1] => 8
[4,2,5,1,3,6] => 5
[4,2,5,1,6,3] => 10
[4,2,5,3,1,6] => 6
[4,2,5,3,6,1] => 11
[4,2,5,6,1,3] => 9
[4,2,5,6,3,1] => 10
[4,2,6,1,3,5] => 5
[4,2,6,1,5,3] => 6
[4,2,6,3,1,5] => 6
[4,2,6,3,5,1] => 7
[4,2,6,5,1,3] => 10
[4,2,6,5,3,1] => 11
[4,3,1,2,5,6] => 4
[4,3,1,2,6,5] => 9
[4,3,1,5,2,6] => 8
[4,3,1,5,6,2] => 13
[4,3,1,6,2,5] => 8
[4,3,1,6,5,2] => 9
[4,3,2,1,5,6] => 5
[4,3,2,1,6,5] => 10
[4,3,2,5,1,6] => 9
[4,3,2,5,6,1] => 14
[4,3,2,6,1,5] => 9
[4,3,2,6,5,1] => 10
[4,3,5,1,2,6] => 7
[4,3,5,1,6,2] => 12
[4,3,5,2,1,6] => 8
[4,3,5,2,6,1] => 13
[4,3,5,6,1,2] => 11
[4,3,5,6,2,1] => 12
[4,3,6,1,2,5] => 7
[4,3,6,1,5,2] => 8
[4,3,6,2,1,5] => 8
[4,3,6,2,5,1] => 9
[4,3,6,5,1,2] => 12
[4,3,6,5,2,1] => 13
[4,5,1,2,3,6] => 3
[4,5,1,2,6,3] => 8
[4,5,1,3,2,6] => 4
[4,5,1,3,6,2] => 9
[4,5,1,6,2,3] => 7
[4,5,1,6,3,2] => 8
[4,5,2,1,3,6] => 4
[4,5,2,1,6,3] => 9
[4,5,2,3,1,6] => 5
[4,5,2,3,6,1] => 10
[4,5,2,6,1,3] => 8
[4,5,2,6,3,1] => 9
[4,5,3,1,2,6] => 5
[4,5,3,1,6,2] => 10
[4,5,3,2,1,6] => 6
[4,5,3,2,6,1] => 11
[4,5,3,6,1,2] => 9
[4,5,3,6,2,1] => 10
[4,5,6,1,2,3] => 6
[4,5,6,1,3,2] => 7
[4,5,6,2,1,3] => 7
[4,5,6,2,3,1] => 8
[4,5,6,3,1,2] => 8
[4,5,6,3,2,1] => 9
[4,6,1,2,3,5] => 3
[4,6,1,2,5,3] => 4
[4,6,1,3,2,5] => 4
[4,6,1,3,5,2] => 5
[4,6,1,5,2,3] => 8
[4,6,1,5,3,2] => 9
[4,6,2,1,3,5] => 4
[4,6,2,1,5,3] => 5
[4,6,2,3,1,5] => 5
[4,6,2,3,5,1] => 6
[4,6,2,5,1,3] => 9
[4,6,2,5,3,1] => 10
[4,6,3,1,2,5] => 5
[4,6,3,1,5,2] => 6
[4,6,3,2,1,5] => 6
[4,6,3,2,5,1] => 7
[4,6,3,5,1,2] => 10
[4,6,3,5,2,1] => 11
[4,6,5,1,2,3] => 7
[4,6,5,1,3,2] => 8
[4,6,5,2,1,3] => 8
[4,6,5,2,3,1] => 9
[4,6,5,3,1,2] => 9
[4,6,5,3,2,1] => 10
[5,1,2,3,4,6] => 1
[5,1,2,3,6,4] => 6
[5,1,2,4,3,6] => 2
[5,1,2,4,6,3] => 7
[5,1,2,6,3,4] => 5
[5,1,2,6,4,3] => 6
[5,1,3,2,4,6] => 2
[5,1,3,2,6,4] => 7
[5,1,3,4,2,6] => 3
[5,1,3,4,6,2] => 8
[5,1,3,6,2,4] => 6
[5,1,3,6,4,2] => 7
[5,1,4,2,3,6] => 5
[5,1,4,2,6,3] => 10
[5,1,4,3,2,6] => 6
[5,1,4,3,6,2] => 11
[5,1,4,6,2,3] => 9
[5,1,4,6,3,2] => 10
[5,1,6,2,3,4] => 4
[5,1,6,2,4,3] => 5
[5,1,6,3,2,4] => 5
[5,1,6,3,4,2] => 6
[5,1,6,4,2,3] => 6
[5,1,6,4,3,2] => 7
[5,2,1,3,4,6] => 2
[5,2,1,3,6,4] => 7
[5,2,1,4,3,6] => 3
[5,2,1,4,6,3] => 8
[5,2,1,6,3,4] => 6
[5,2,1,6,4,3] => 7
[5,2,3,1,4,6] => 3
[5,2,3,1,6,4] => 8
[5,2,3,4,1,6] => 4
[5,2,3,4,6,1] => 9
[5,2,3,6,1,4] => 7
[5,2,3,6,4,1] => 8
[5,2,4,1,3,6] => 6
[5,2,4,1,6,3] => 11
[5,2,4,3,1,6] => 7
[5,2,4,3,6,1] => 12
[5,2,4,6,1,3] => 10
[5,2,4,6,3,1] => 11
[5,2,6,1,3,4] => 5
[5,2,6,1,4,3] => 6
[5,2,6,3,1,4] => 6
[5,2,6,3,4,1] => 7
[5,2,6,4,1,3] => 7
[5,2,6,4,3,1] => 8
[5,3,1,2,4,6] => 4
[5,3,1,2,6,4] => 9
[5,3,1,4,2,6] => 5
[5,3,1,4,6,2] => 10
[5,3,1,6,2,4] => 8
[5,3,1,6,4,2] => 9
[5,3,2,1,4,6] => 5
[5,3,2,1,6,4] => 10
[5,3,2,4,1,6] => 6
[5,3,2,4,6,1] => 11
[5,3,2,6,1,4] => 9
[5,3,2,6,4,1] => 10
[5,3,4,1,2,6] => 8
[5,3,4,1,6,2] => 13
[5,3,4,2,1,6] => 9
[5,3,4,2,6,1] => 14
[5,3,4,6,1,2] => 12
[5,3,4,6,2,1] => 13
[5,3,6,1,2,4] => 7
[5,3,6,1,4,2] => 8
[5,3,6,2,1,4] => 8
[5,3,6,2,4,1] => 9
[5,3,6,4,1,2] => 9
[5,3,6,4,2,1] => 10
[5,4,1,2,3,6] => 4
[5,4,1,2,6,3] => 9
[5,4,1,3,2,6] => 5
[5,4,1,3,6,2] => 10
[5,4,1,6,2,3] => 8
[5,4,1,6,3,2] => 9
[5,4,2,1,3,6] => 5
[5,4,2,1,6,3] => 10
[5,4,2,3,1,6] => 6
[5,4,2,3,6,1] => 11
[5,4,2,6,1,3] => 9
[5,4,2,6,3,1] => 10
[5,4,3,1,2,6] => 6
[5,4,3,1,6,2] => 11
[5,4,3,2,1,6] => 7
[5,4,3,2,6,1] => 12
[5,4,3,6,1,2] => 10
[5,4,3,6,2,1] => 11
[5,4,6,1,2,3] => 7
[5,4,6,1,3,2] => 8
[5,4,6,2,1,3] => 8
[5,4,6,2,3,1] => 9
[5,4,6,3,1,2] => 9
[5,4,6,3,2,1] => 10
[5,6,1,2,3,4] => 3
[5,6,1,2,4,3] => 4
[5,6,1,3,2,4] => 4
[5,6,1,3,4,2] => 5
[5,6,1,4,2,3] => 5
[5,6,1,4,3,2] => 6
[5,6,2,1,3,4] => 4
[5,6,2,1,4,3] => 5
[5,6,2,3,1,4] => 5
[5,6,2,3,4,1] => 6
[5,6,2,4,1,3] => 6
[5,6,2,4,3,1] => 7
[5,6,3,1,2,4] => 5
[5,6,3,1,4,2] => 6
[5,6,3,2,1,4] => 6
[5,6,3,2,4,1] => 7
[5,6,3,4,1,2] => 7
[5,6,3,4,2,1] => 8
[5,6,4,1,2,3] => 8
[5,6,4,1,3,2] => 9
[5,6,4,2,1,3] => 9
[5,6,4,2,3,1] => 10
[5,6,4,3,1,2] => 10
[5,6,4,3,2,1] => 11
[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] => 6
[6,1,2,5,4,3] => 7
[6,1,3,2,4,5] => 2
[6,1,3,2,5,4] => 3
[6,1,3,4,2,5] => 3
[6,1,3,4,5,2] => 4
[6,1,3,5,2,4] => 7
[6,1,3,5,4,2] => 8
[6,1,4,2,3,5] => 5
[6,1,4,2,5,3] => 6
[6,1,4,3,2,5] => 6
[6,1,4,3,5,2] => 7
[6,1,4,5,2,3] => 10
[6,1,4,5,3,2] => 11
[6,1,5,2,3,4] => 5
[6,1,5,2,4,3] => 6
[6,1,5,3,2,4] => 6
[6,1,5,3,4,2] => 7
[6,1,5,4,2,3] => 7
[6,1,5,4,3,2] => 8
[6,2,1,3,4,5] => 2
[6,2,1,3,5,4] => 3
[6,2,1,4,3,5] => 3
[6,2,1,4,5,3] => 4
[6,2,1,5,3,4] => 7
[6,2,1,5,4,3] => 8
[6,2,3,1,4,5] => 3
[6,2,3,1,5,4] => 4
[6,2,3,4,1,5] => 4
[6,2,3,4,5,1] => 5
[6,2,3,5,1,4] => 8
[6,2,3,5,4,1] => 9
[6,2,4,1,3,5] => 6
[6,2,4,1,5,3] => 7
[6,2,4,3,1,5] => 7
[6,2,4,3,5,1] => 8
[6,2,4,5,1,3] => 11
[6,2,4,5,3,1] => 12
[6,2,5,1,3,4] => 6
[6,2,5,1,4,3] => 7
[6,2,5,3,1,4] => 7
[6,2,5,3,4,1] => 8
[6,2,5,4,1,3] => 8
[6,2,5,4,3,1] => 9
[6,3,1,2,4,5] => 4
[6,3,1,2,5,4] => 5
[6,3,1,4,2,5] => 5
[6,3,1,4,5,2] => 6
[6,3,1,5,2,4] => 9
[6,3,1,5,4,2] => 10
[6,3,2,1,4,5] => 5
[6,3,2,1,5,4] => 6
[6,3,2,4,1,5] => 6
[6,3,2,4,5,1] => 7
[6,3,2,5,1,4] => 10
[6,3,2,5,4,1] => 11
[6,3,4,1,2,5] => 8
[6,3,4,1,5,2] => 9
[6,3,4,2,1,5] => 9
[6,3,4,2,5,1] => 10
[6,3,4,5,1,2] => 13
[6,3,4,5,2,1] => 14
[6,3,5,1,2,4] => 8
[6,3,5,1,4,2] => 9
[6,3,5,2,1,4] => 9
[6,3,5,2,4,1] => 10
[6,3,5,4,1,2] => 10
[6,3,5,4,2,1] => 11
[6,4,1,2,3,5] => 4
[6,4,1,2,5,3] => 5
[6,4,1,3,2,5] => 5
[6,4,1,3,5,2] => 6
[6,4,1,5,2,3] => 9
[6,4,1,5,3,2] => 10
[6,4,2,1,3,5] => 5
[6,4,2,1,5,3] => 6
[6,4,2,3,1,5] => 6
[6,4,2,3,5,1] => 7
[6,4,2,5,1,3] => 10
[6,4,2,5,3,1] => 11
[6,4,3,1,2,5] => 6
[6,4,3,1,5,2] => 7
[6,4,3,2,1,5] => 7
[6,4,3,2,5,1] => 8
[6,4,3,5,1,2] => 11
[6,4,3,5,2,1] => 12
[6,4,5,1,2,3] => 8
[6,4,5,1,3,2] => 9
[6,4,5,2,1,3] => 9
[6,4,5,2,3,1] => 10
[6,4,5,3,1,2] => 10
[6,4,5,3,2,1] => 11
[6,5,1,2,3,4] => 4
[6,5,1,2,4,3] => 5
[6,5,1,3,2,4] => 5
[6,5,1,3,4,2] => 6
[6,5,1,4,2,3] => 6
[6,5,1,4,3,2] => 7
[6,5,2,1,3,4] => 5
[6,5,2,1,4,3] => 6
[6,5,2,3,1,4] => 6
[6,5,2,3,4,1] => 7
[6,5,2,4,1,3] => 7
[6,5,2,4,3,1] => 8
[6,5,3,1,2,4] => 6
[6,5,3,1,4,2] => 7
[6,5,3,2,1,4] => 7
[6,5,3,2,4,1] => 8
[6,5,3,4,1,2] => 8
[6,5,3,4,2,1] => 9
[6,5,4,1,2,3] => 9
[6,5,4,1,3,2] => 10
[6,5,4,2,1,3] => 10
[6,5,4,2,3,1] => 11
[6,5,4,3,1,2] => 11
[6,5,4,3,2,1] => 12
-----------------------------------------------------------------------------
Created: Jul 24, 2013 at 12:25 by Christian Stump
-----------------------------------------------------------------------------
Last Updated: Feb 27, 2023 at 12:37 by Martin Rubey