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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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-----------------------------------------------------------------------------
Statistic identifier: St001822
-----------------------------------------------------------------------------
Collection: Signed permutations
-----------------------------------------------------------------------------
Description: The number of alignments of a signed permutation.
An alignment of a signed permutation $n\in\mathfrak H_n$ is either a nesting alignment, [[St001866]], an alignment of type EN, [[St001867]], or an alignment of type NE, [[St001868]].
Let $\operatorname{al}$ be the number of alignments of $\pi$, let \operatorname{cr} be the number of crossings, [[St001862]], let \operatorname{wex} be the number of weak excedances, [[St001863]], and let \operatorname{neg} be the number of negative entries, [[St001429]]. Then, $\operatorname{al}+\operatorname{cr}=(n-\operatorname{wex})(\operatorname{wex}-1+\operatorname{neg})+\binom{\operatorname{neg}{2}$.
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References: [1] Cho, S., Park, K. Permutation statistics and weak Bruhat order in permutation tableaux of type $B$ [[MathSciNet:3319075]]
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Code:
def statistic(pi):
n = len(pi)
A_nest = [(i, j) for i in range(1, n+1) for j in range(1, n+1)
if (-i < -j < -pi(j) < -pi(i))
or (-i < j <= pi(j) < -pi(i))
or (i < j <= pi(j) < pi(i))]
A_EN = [(i, j) for i in range(1, n+1) for j in range(1, n+1)
if (-i < 0 < -pi(i) < pi(j) < j)
or (i <= pi(i) < pi(j) < j)]
A_NE = [(i, j) for i in range(1, n+1) for j in range(1, n+1)
if (pi(i) < i < j <= pi(j))]
return len(A_nest + A_EN + A_NE)
-----------------------------------------------------------------------------
Statistic values:
[1] => 0
[-1] => 0
[1,2] => 0
[1,-2] => 1
[-1,2] => 1
[-1,-2] => 1
[2,1] => 0
[2,-1] => 0
[-2,1] => 0
[-2,-1] => 0
[1,2,3] => 0
[1,2,-3] => 2
[1,-2,3] => 2
[1,-2,-3] => 3
[-1,2,3] => 2
[-1,2,-3] => 3
[-1,-2,3] => 3
[-1,-2,-3] => 3
[1,3,2] => 1
[1,3,-2] => 1
[1,-3,2] => 2
[1,-3,-2] => 2
[-1,3,2] => 2
[-1,3,-2] => 2
[-1,-3,2] => 2
[-1,-3,-2] => 2
[2,1,3] => 1
[2,1,-3] => 2
[2,-1,3] => 1
[2,-1,-3] => 2
[-2,1,3] => 2
[-2,1,-3] => 2
[-2,-1,3] => 2
[-2,-1,-3] => 2
[2,3,1] => 0
[2,3,-1] => 0
[2,-3,1] => 1
[2,-3,-1] => 1
[-2,3,1] => 1
[-2,3,-1] => 1
[-2,-3,1] => 1
[-2,-3,-1] => 1
[3,1,2] => 0
[3,1,-2] => 1
[3,-1,2] => 1
[3,-1,-2] => 1
[-3,1,2] => 0
[-3,1,-2] => 1
[-3,-1,2] => 1
[-3,-1,-2] => 1
[3,2,1] => 1
[3,2,-1] => 1
[3,-2,1] => 0
[3,-2,-1] => 0
[-3,2,1] => 2
[-3,2,-1] => 2
[-3,-2,1] => 0
[-3,-2,-1] => 0
[1,2,3,4] => 0
[1,2,3,-4] => 3
[1,2,-3,4] => 3
[1,2,-3,-4] => 5
[1,-2,3,4] => 3
[1,-2,3,-4] => 5
[1,-2,-3,4] => 5
[1,-2,-3,-4] => 6
[-1,2,3,4] => 3
[-1,2,3,-4] => 5
[-1,2,-3,4] => 5
[-1,2,-3,-4] => 6
[-1,-2,3,4] => 5
[-1,-2,3,-4] => 6
[-1,-2,-3,4] => 6
[-1,-2,-3,-4] => 6
[1,2,4,3] => 2
[1,2,4,-3] => 2
[1,2,-4,3] => 4
[1,2,-4,-3] => 4
[1,-2,4,3] => 4
[1,-2,4,-3] => 4
[1,-2,-4,3] => 5
[1,-2,-4,-3] => 5
[-1,2,4,3] => 4
[-1,2,4,-3] => 4
[-1,2,-4,3] => 5
[-1,2,-4,-3] => 5
[-1,-2,4,3] => 5
[-1,-2,4,-3] => 5
[-1,-2,-4,3] => 5
[-1,-2,-4,-3] => 5
[1,3,2,4] => 2
[1,3,2,-4] => 4
[1,3,-2,4] => 2
[1,3,-2,-4] => 4
[1,-3,2,4] => 4
[1,-3,2,-4] => 5
[1,-3,-2,4] => 4
[1,-3,-2,-4] => 5
[-1,3,2,4] => 4
[-1,3,2,-4] => 5
[-1,3,-2,4] => 4
[-1,3,-2,-4] => 5
[-1,-3,2,4] => 5
[-1,-3,2,-4] => 5
[-1,-3,-2,4] => 5
[-1,-3,-2,-4] => 5
[1,3,4,2] => 1
[1,3,4,-2] => 1
[1,3,-4,2] => 3
[1,3,-4,-2] => 3
[1,-3,4,2] => 3
[1,-3,4,-2] => 3
[1,-3,-4,2] => 4
[1,-3,-4,-2] => 4
[-1,3,4,2] => 3
[-1,3,4,-2] => 3
[-1,3,-4,2] => 4
[-1,3,-4,-2] => 4
[-1,-3,4,2] => 4
[-1,-3,4,-2] => 4
[-1,-3,-4,2] => 4
[-1,-3,-4,-2] => 4
[1,4,2,3] => 2
[1,4,2,-3] => 3
[1,4,-2,3] => 3
[1,4,-2,-3] => 3
[1,-4,2,3] => 3
[1,-4,2,-3] => 4
[1,-4,-2,3] => 4
[1,-4,-2,-3] => 4
[-1,4,2,3] => 3
[-1,4,2,-3] => 4
[-1,4,-2,3] => 4
[-1,4,-2,-3] => 4
[-1,-4,2,3] => 3
[-1,-4,2,-3] => 4
[-1,-4,-2,3] => 4
[-1,-4,-2,-3] => 4
[1,4,3,2] => 2
[1,4,3,-2] => 2
[1,4,-3,2] => 2
[1,4,-3,-2] => 2
[1,-4,3,2] => 4
[1,-4,3,-2] => 4
[1,-4,-3,2] => 3
[1,-4,-3,-2] => 3
[-1,4,3,2] => 4
[-1,4,3,-2] => 4
[-1,4,-3,2] => 3
[-1,4,-3,-2] => 3
[-1,-4,3,2] => 5
[-1,-4,3,-2] => 5
[-1,-4,-3,2] => 3
[-1,-4,-3,-2] => 3
[2,1,3,4] => 2
[2,1,3,-4] => 4
[2,1,-3,4] => 4
[2,1,-3,-4] => 5
[2,-1,3,4] => 2
[2,-1,3,-4] => 4
[2,-1,-3,4] => 4
[2,-1,-3,-4] => 5
[-2,1,3,4] => 4
[-2,1,3,-4] => 5
[-2,1,-3,4] => 5
[-2,1,-3,-4] => 5
[-2,-1,3,4] => 4
[-2,-1,3,-4] => 5
[-2,-1,-3,4] => 5
[-2,-1,-3,-4] => 5
[2,1,4,3] => 2
[2,1,4,-3] => 3
[2,1,-4,3] => 3
[2,1,-4,-3] => 4
[2,-1,4,3] => 3
[2,-1,4,-3] => 3
[2,-1,-4,3] => 4
[2,-1,-4,-3] => 4
[-2,1,4,3] => 3
[-2,1,4,-3] => 4
[-2,1,-4,3] => 3
[-2,1,-4,-3] => 4
[-2,-1,4,3] => 4
[-2,-1,4,-3] => 4
[-2,-1,-4,3] => 4
[-2,-1,-4,-3] => 4
[2,3,1,4] => 1
[2,3,1,-4] => 3
[2,3,-1,4] => 1
[2,3,-1,-4] => 3
[2,-3,1,4] => 3
[2,-3,1,-4] => 4
[2,-3,-1,4] => 3
[2,-3,-1,-4] => 4
[-2,3,1,4] => 3
[-2,3,1,-4] => 4
[-2,3,-1,4] => 3
[-2,3,-1,-4] => 4
[-2,-3,1,4] => 4
[-2,-3,1,-4] => 4
[-2,-3,-1,4] => 4
[-2,-3,-1,-4] => 4
[2,3,4,1] => 0
[2,3,4,-1] => 0
[2,3,-4,1] => 2
[2,3,-4,-1] => 2
[2,-3,4,1] => 2
[2,-3,4,-1] => 2
[2,-3,-4,1] => 3
[2,-3,-4,-1] => 3
[-2,3,4,1] => 2
[-2,3,4,-1] => 2
[-2,3,-4,1] => 3
[-2,3,-4,-1] => 3
[-2,-3,4,1] => 3
[-2,-3,4,-1] => 3
[-2,-3,-4,1] => 3
[-2,-3,-4,-1] => 3
[2,4,1,3] => 1
[2,4,1,-3] => 2
[2,4,-1,3] => 2
[2,4,-1,-3] => 2
[2,-4,1,3] => 2
[2,-4,1,-3] => 3
[2,-4,-1,3] => 3
[2,-4,-1,-3] => 3
[-2,4,1,3] => 2
[-2,4,1,-3] => 3
[-2,4,-1,3] => 3
[-2,4,-1,-3] => 3
[-2,-4,1,3] => 2
[-2,-4,1,-3] => 3
[-2,-4,-1,3] => 3
[-2,-4,-1,-3] => 3
[2,4,3,1] => 1
[2,4,3,-1] => 1
[2,4,-3,1] => 1
[2,4,-3,-1] => 1
[2,-4,3,1] => 3
[2,-4,3,-1] => 3
[2,-4,-3,1] => 2
[2,-4,-3,-1] => 2
[-2,4,3,1] => 3
[-2,4,3,-1] => 3
[-2,4,-3,1] => 2
[-2,4,-3,-1] => 2
[-2,-4,3,1] => 4
[-2,-4,3,-1] => 4
[-2,-4,-3,1] => 2
[-2,-4,-3,-1] => 2
[3,1,2,4] => 2
[3,1,2,-4] => 3
[3,1,-2,4] => 3
[3,1,-2,-4] => 4
[3,-1,2,4] => 3
[3,-1,2,-4] => 4
[3,-1,-2,4] => 3
[3,-1,-2,-4] => 4
[-3,1,2,4] => 3
[-3,1,2,-4] => 3
[-3,1,-2,4] => 4
[-3,1,-2,-4] => 4
[-3,-1,2,4] => 4
[-3,-1,2,-4] => 4
[-3,-1,-2,4] => 4
[-3,-1,-2,-4] => 4
[3,1,4,2] => 1
[3,1,4,-2] => 2
[3,1,-4,2] => 2
[3,1,-4,-2] => 3
[3,-1,4,2] => 2
[3,-1,4,-2] => 2
[3,-1,-4,2] => 3
[3,-1,-4,-2] => 3
[-3,1,4,2] => 2
[-3,1,4,-2] => 3
[-3,1,-4,2] => 2
[-3,1,-4,-2] => 3
[-3,-1,4,2] => 3
[-3,-1,4,-2] => 3
[-3,-1,-4,2] => 3
[-3,-1,-4,-2] => 3
[3,2,1,4] => 2
[3,2,1,-4] => 4
[3,2,-1,4] => 2
[3,2,-1,-4] => 4
[3,-2,1,4] => 2
[3,-2,1,-4] => 3
[3,-2,-1,4] => 2
[3,-2,-1,-4] => 3
[-3,2,1,4] => 4
[-3,2,1,-4] => 5
[-3,2,-1,4] => 4
[-3,2,-1,-4] => 5
[-3,-2,1,4] => 3
[-3,-2,1,-4] => 3
[-3,-2,-1,4] => 3
[-3,-2,-1,-4] => 3
[3,2,4,1] => 1
[3,2,4,-1] => 1
[3,2,-4,1] => 3
[3,2,-4,-1] => 3
[3,-2,4,1] => 1
[3,-2,4,-1] => 1
[3,-2,-4,1] => 2
[3,-2,-4,-1] => 2
[-3,2,4,1] => 3
[-3,2,4,-1] => 3
[-3,2,-4,1] => 4
[-3,2,-4,-1] => 4
[-3,-2,4,1] => 2
[-3,-2,4,-1] => 2
[-3,-2,-4,1] => 2
[-3,-2,-4,-1] => 2
[3,4,1,2] => 0
[3,4,1,-2] => 1
[3,4,-1,2] => 1
[3,4,-1,-2] => 1
[3,-4,1,2] => 1
[3,-4,1,-2] => 2
[3,-4,-1,2] => 2
[3,-4,-1,-2] => 2
[-3,4,1,2] => 1
[-3,4,1,-2] => 2
[-3,4,-1,2] => 2
[-3,4,-1,-2] => 2
[-3,-4,1,2] => 1
[-3,-4,1,-2] => 2
[-3,-4,-1,2] => 2
[-3,-4,-1,-2] => 2
[3,4,2,1] => 1
[3,4,2,-1] => 2
[3,4,-2,1] => 0
[3,4,-2,-1] => 0
[3,-4,2,1] => 2
[3,-4,2,-1] => 3
[3,-4,-2,1] => 1
[3,-4,-2,-1] => 1
[-3,4,2,1] => 2
[-3,4,2,-1] => 3
[-3,4,-2,1] => 1
[-3,4,-2,-1] => 1
[-3,-4,2,1] => 2
[-3,-4,2,-1] => 3
[-3,-4,-2,1] => 1
[-3,-4,-2,-1] => 1
[4,1,2,3] => 0
[4,1,2,-3] => 2
[4,1,-2,3] => 2
[4,1,-2,-3] => 3
[4,-1,2,3] => 2
[4,-1,2,-3] => 3
[4,-1,-2,3] => 3
[4,-1,-2,-3] => 3
[-4,1,2,3] => 0
[-4,1,2,-3] => 2
[-4,1,-2,3] => 2
[-4,1,-2,-3] => 3
[-4,-1,2,3] => 2
[-4,-1,2,-3] => 3
[-4,-1,-2,3] => 3
[-4,-1,-2,-3] => 3
[4,1,3,2] => 2
[4,1,3,-2] => 3
[4,1,-3,2] => 1
[4,1,-3,-2] => 2
[4,-1,3,2] => 3
[4,-1,3,-2] => 3
[4,-1,-3,2] => 2
[4,-1,-3,-2] => 2
[-4,1,3,2] => 3
[-4,1,3,-2] => 4
[-4,1,-3,2] => 1
[-4,1,-3,-2] => 2
[-4,-1,3,2] => 4
[-4,-1,3,-2] => 4
[-4,-1,-3,2] => 2
[-4,-1,-3,-2] => 2
[4,2,1,3] => 2
[4,2,1,-3] => 3
[4,2,-1,3] => 3
[4,2,-1,-3] => 3
[4,-2,1,3] => 1
[4,-2,1,-3] => 2
[4,-2,-1,3] => 2
[4,-2,-1,-3] => 2
[-4,2,1,3] => 3
[-4,2,1,-3] => 4
[-4,2,-1,3] => 4
[-4,2,-1,-3] => 4
[-4,-2,1,3] => 1
[-4,-2,1,-3] => 2
[-4,-2,-1,3] => 2
[-4,-2,-1,-3] => 2
[4,2,3,1] => 2
[4,2,3,-1] => 2
[4,2,-3,1] => 2
[4,2,-3,-1] => 2
[4,-2,3,1] => 2
[4,-2,3,-1] => 2
[4,-2,-3,1] => 1
[4,-2,-3,-1] => 1
[-4,2,3,1] => 4
[-4,2,3,-1] => 4
[-4,2,-3,1] => 3
[-4,2,-3,-1] => 3
[-4,-2,3,1] => 3
[-4,-2,3,-1] => 3
[-4,-2,-3,1] => 1
[-4,-2,-3,-1] => 1
[4,3,1,2] => 1
[4,3,1,-2] => 2
[4,3,-1,2] => 2
[4,3,-1,-2] => 2
[4,-3,1,2] => 0
[4,-3,1,-2] => 1
[4,-3,-1,2] => 1
[4,-3,-1,-2] => 1
[-4,3,1,2] => 2
[-4,3,1,-2] => 3
[-4,3,-1,2] => 3
[-4,3,-1,-2] => 3
[-4,-3,1,2] => 0
[-4,-3,1,-2] => 1
[-4,-3,-1,2] => 1
[-4,-3,-1,-2] => 1
[4,3,2,1] => 2
[4,3,2,-1] => 3
[4,3,-2,1] => 1
[4,3,-2,-1] => 1
[4,-3,2,1] => 1
[4,-3,2,-1] => 2
[4,-3,-2,1] => 0
[4,-3,-2,-1] => 0
[-4,3,2,1] => 3
[-4,3,2,-1] => 4
[-4,3,-2,1] => 2
[-4,3,-2,-1] => 2
[-4,-3,2,1] => 1
[-4,-3,2,-1] => 2
[-4,-3,-2,1] => 0
[-4,-3,-2,-1] => 0
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Created: Jul 21, 2022 at 07:29 by Dennis Jahn
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Last Updated: Dec 01, 2022 at 14:23 by Martin Rubey