Identifier
Values
[1] => 0
[-1] => 0
[1,2] => 0
[1,-2] => 1
[-1,2] => 0
[-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] => 1
[1,-2,-3] => 3
[-1,2,3] => 0
[-1,2,-3] => 2
[-1,-2,3] => 1
[-1,-2,-3] => 3
[1,3,2] => 0
[1,3,-2] => 1
[1,-3,2] => 1
[1,-3,-2] => 2
[-1,3,2] => 0
[-1,3,-2] => 1
[-1,-3,2] => 1
[-1,-3,-2] => 2
[2,1,3] => 0
[2,1,-3] => 2
[2,-1,3] => 0
[2,-1,-3] => 2
[-2,1,3] => 0
[-2,1,-3] => 2
[-2,-1,3] => 0
[-2,-1,-3] => 2
[2,3,1] => 0
[2,3,-1] => 0
[2,-3,1] => 1
[2,-3,-1] => 1
[-2,3,1] => 0
[-2,3,-1] => 0
[-2,-3,1] => 1
[-2,-3,-1] => 1
[3,1,2] => 0
[3,1,-2] => 1
[3,-1,2] => 0
[3,-1,-2] => 1
[-3,1,2] => 0
[-3,1,-2] => 1
[-3,-1,2] => 0
[-3,-1,-2] => 1
[3,2,1] => 1
[3,2,-1] => 1
[3,-2,1] => 0
[3,-2,-1] => 0
[-3,2,1] => 1
[-3,2,-1] => 1
[-3,-2,1] => 0
[-3,-2,-1] => 0
[1,2,3,4] => 0
[1,2,3,-4] => 3
[1,2,-3,4] => 2
[1,2,-3,-4] => 5
[1,-2,3,4] => 1
[1,-2,3,-4] => 4
[1,-2,-3,4] => 3
[1,-2,-3,-4] => 6
[-1,2,3,4] => 0
[-1,2,3,-4] => 3
[-1,2,-3,4] => 2
[-1,2,-3,-4] => 5
[-1,-2,3,4] => 1
[-1,-2,3,-4] => 4
[-1,-2,-3,4] => 3
[-1,-2,-3,-4] => 6
[1,2,4,3] => 0
[1,2,4,-3] => 2
[1,2,-4,3] => 2
[1,2,-4,-3] => 4
[1,-2,4,3] => 1
[1,-2,4,-3] => 3
[1,-2,-4,3] => 3
[1,-2,-4,-3] => 5
[-1,2,4,3] => 0
[-1,2,4,-3] => 2
[-1,2,-4,3] => 2
[-1,2,-4,-3] => 4
[-1,-2,4,3] => 1
[-1,-2,4,-3] => 3
[-1,-2,-4,3] => 3
[-1,-2,-4,-3] => 5
[1,3,2,4] => 0
[1,3,2,-4] => 3
[1,3,-2,4] => 1
[1,3,-2,-4] => 4
[1,-3,2,4] => 1
[1,-3,2,-4] => 4
[1,-3,-2,4] => 2
[1,-3,-2,-4] => 5
[-1,3,2,4] => 0
[-1,3,2,-4] => 3
[-1,3,-2,4] => 1
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Description
The nesting alignments of a signed permutation.
A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that
A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that
- $-i < -j < -\pi(j) < -\pi(i)$, or
- $-i < j \leq \pi(j) < -\pi(i)$, or
- $i < j \leq \pi(j) < \pi(i)$.
References
[1] Cho, S., Park, K. Permutation statistics and weak \B\ruhat order in permutation tableaux of type $B$ arXiv:1412.6290
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))]
return len(A_nest)
Created
Dec 01, 2022 at 11:28 by Martin Rubey
Updated
Dec 01, 2022 at 11:28 by Martin Rubey
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