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Matching statistic: St001865
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St001865: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[-1] => 0
[1,2] => 4
[1,-2] => 3
[-1,2] => 3
[-1,-2] => 2
[2,1] => 2
[2,-1] => 1
[-2,1] => 1
[-2,-1] => 0
[1,2,3] => 9
[1,2,-3] => 8
[1,-2,3] => 8
[1,-2,-3] => 7
[-1,2,3] => 8
[-1,2,-3] => 7
[-1,-2,3] => 7
[-1,-2,-3] => 6
[1,3,2] => 7
[1,3,-2] => 6
[1,-3,2] => 6
[1,-3,-2] => 5
[-1,3,2] => 6
[-1,3,-2] => 5
[-1,-3,2] => 5
[-1,-3,-2] => 4
[2,1,3] => 7
[2,1,-3] => 6
[2,-1,3] => 6
[2,-1,-3] => 5
[-2,1,3] => 6
[-2,1,-3] => 5
[-2,-1,3] => 5
[-2,-1,-3] => 4
[2,3,1] => 5
[2,3,-1] => 4
[2,-3,1] => 4
[2,-3,-1] => 3
[-2,3,1] => 4
[-2,3,-1] => 3
[-2,-3,1] => 3
[-2,-3,-1] => 2
[3,1,2] => 5
[3,1,-2] => 4
[3,-1,2] => 4
[3,-1,-2] => 3
[-3,1,2] => 4
[-3,1,-2] => 3
[-3,-1,2] => 3
[-3,-1,-2] => 2
Description
The number of alignments of a signed permutation.
An alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $-n \leq i \leq j \leq n$, $i,j\neq 0$, such that one of the following conditions hold:
* $i < j \leq \pi(j) < \pi(i)$, and $j > 0$ if it is a fixed point, or
* $\pi(j) < \pi(i) \leq i < j)$ and $i < 0$ if it is a fixed point, or
* $i \leq \pi(i) < \pi(j) \leq j$ and $i > 0$ if it is a fixed point and $j < 0$ if it is a fixed point, or
* $\pi(i) \leq i < j \leq \pi(j)$ and $i < 0$ if it is a fixed point and $j > 0$ if it is a fixed point.
Let $al$ be the number of alignments of $\pi$, $cr$ be the number of crossings, [[St001862]], and let $fwex$ be the number of flag weak excedances, [[St001817]]. Then
$$2 cr + al = n^2 - 2n + fwex.$$
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