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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 π∈Hn is a pair −n≤i≤j≤n, i,j≠0, such that one of the following conditions hold:
* i<j≤π(j)<π(i), and j>0 if it is a fixed point, or
* π(j)<π(i)≤i<j) and i<0 if it is a fixed point, or
* i≤π(i)<π(j)≤j and i>0 if it is a fixed point and j<0 if it is a fixed point, or
* π(i)≤i<j≤π(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 π, cr be the number of crossings, [[St001862]], and let fwex be the number of flag weak excedances, [[St001817]]. Then
2cr+al=n2−2n+fwex.
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