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Matching statistic: St001821
St001821: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[-1] => 1
[1,2] => 0
[1,-2] => 3
[-1,2] => 1
[-1,-2] => 4
[2,1] => 1
[2,-1] => 2
[-2,1] => 3
[-2,-1] => 2
[1,2,3] => 0
[1,2,-3] => 5
[1,-2,3] => 3
[1,-2,-3] => 8
[-1,2,3] => 1
[-1,2,-3] => 6
[-1,-2,3] => 4
[-1,-2,-3] => 9
[1,3,2] => 1
[1,3,-2] => 4
[1,-3,2] => 7
[1,-3,-2] => 4
[-1,3,2] => 2
[-1,3,-2] => 5
[-1,-3,2] => 8
[-1,-3,-2] => 5
[2,1,3] => 1
[2,1,-3] => 6
[2,-1,3] => 2
[2,-1,-3] => 7
[-2,1,3] => 3
[-2,1,-3] => 8
[-2,-1,3] => 2
[-2,-1,-3] => 7
[2,3,1] => 2
[2,3,-1] => 3
[2,-3,1] => 6
[2,-3,-1] => 5
[-2,3,1] => 4
[-2,3,-1] => 3
[-2,-3,1] => 6
[-2,-3,-1] => 7
[3,1,2] => 3
[3,1,-2] => 5
[3,-1,2] => 4
[3,-1,-2] => 4
[-3,1,2] => 6
[-3,1,-2] => 4
[-3,-1,2] => 5
[-3,-1,-2] => 5
Description
The sorting index of a signed permutation.
A signed permutation σ=[σ(1),…,σ(n)] can be sorted [1,…,n] by signed transpositions in the following way:
First move ±n to its position and swap the sign if needed, then ±(n−1),±(n−2) and so on.
For example for [2,−4,5,−1,−3] we have the swaps
[2,−4,5,−1,−3]→[2,−4,−3,−1,5]→[2,1,−3,4,5]→[2,1,3,4,5]→[1,2,3,4,5]
given by the signed transpositions (3,5),(−2,4),(−3,3),(1,2).
If (i1,j1),…,(in,jn) is the decomposition of σ obtained this way (including trivial transpositions) then the sorting index of σ is defined as
sorB(σ)=n−1∑k=1jk−ik−χ(ik<0),
where χ(ik<0) is 1 if ik is negative and 0 otherwise.
For σ=[2,−4,5,−1,−3] we have
sorB(σ)=(5−3)+(4−(−2)−1)+(3−(−3)−1)+(2−1)=13.
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