Identifier
Values
[1] => [1] => [-1] => 0
[1,2] => [1,2] => [-1,-2] => 0
[2,1] => [2,1] => [-2,-1] => 1
[1,2,3] => [1,2,3] => [-1,-2,-3] => 0
[1,3,2] => [1,3,2] => [-1,-3,-2] => 1
[2,1,3] => [2,1,3] => [-2,-1,-3] => 1
[2,3,1] => [2,3,1] => [-2,-3,-1] => 2
[3,1,2] => [3,1,2] => [-3,-1,-2] => 2
[3,2,1] => [3,2,1] => [-3,-2,-1] => 3
[1,2,3,4] => [1,2,3,4] => [-1,-2,-3,-4] => 0
[1,2,4,3] => [1,2,4,3] => [-1,-2,-4,-3] => 1
[1,3,2,4] => [1,3,2,4] => [-1,-3,-2,-4] => 1
[1,3,4,2] => [1,3,4,2] => [-1,-3,-4,-2] => 2
[1,4,2,3] => [1,4,2,3] => [-1,-4,-2,-3] => 2
[1,4,3,2] => [1,4,3,2] => [-1,-4,-3,-2] => 3
[2,1,3,4] => [2,1,3,4] => [-2,-1,-3,-4] => 1
[2,1,4,3] => [2,1,4,3] => [-2,-1,-4,-3] => 2
[2,3,1,4] => [2,3,1,4] => [-2,-3,-1,-4] => 2
[2,3,4,1] => [2,3,4,1] => [-2,-3,-4,-1] => 3
[2,4,1,3] => [2,4,1,3] => [-2,-4,-1,-3] => 3
[2,4,3,1] => [2,4,3,1] => [-2,-4,-3,-1] => 4
[3,1,2,4] => [3,1,2,4] => [-3,-1,-2,-4] => 2
[3,1,4,2] => [3,1,4,2] => [-3,-1,-4,-2] => 3
[3,2,1,4] => [3,2,1,4] => [-3,-2,-1,-4] => 3
[3,2,4,1] => [3,2,4,1] => [-3,-2,-4,-1] => 4
[3,4,1,2] => [3,4,1,2] => [-3,-4,-1,-2] => 4
[3,4,2,1] => [3,4,2,1] => [-3,-4,-2,-1] => 5
[4,1,2,3] => [4,1,2,3] => [-4,-1,-2,-3] => 3
[4,1,3,2] => [4,1,3,2] => [-4,-1,-3,-2] => 4
[4,2,1,3] => [4,2,1,3] => [-4,-2,-1,-3] => 4
[4,2,3,1] => [4,2,3,1] => [-4,-2,-3,-1] => 5
[4,3,1,2] => [4,3,1,2] => [-4,-3,-1,-2] => 5
[4,3,2,1] => [4,3,2,1] => [-4,-3,-2,-1] => 6
[1,2,3,4,5] => [1,2,3,4,5] => [-1,-2,-3,-4,-5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [-1,-2,-3,-5,-4] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [-1,-2,-4,-3,-5] => 1
[1,2,4,5,3] => [1,2,4,5,3] => [-1,-2,-4,-5,-3] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [-1,-2,-5,-3,-4] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [-1,-2,-5,-4,-3] => 3
[1,3,2,4,5] => [1,3,2,4,5] => [-1,-3,-2,-4,-5] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [-1,-3,-2,-5,-4] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [-1,-3,-4,-2,-5] => 2
[1,3,4,5,2] => [1,3,4,5,2] => [-1,-3,-4,-5,-2] => 3
[1,3,5,2,4] => [1,3,5,2,4] => [-1,-3,-5,-2,-4] => 3
[1,3,5,4,2] => [1,3,5,4,2] => [-1,-3,-5,-4,-2] => 4
[1,4,2,3,5] => [1,4,2,3,5] => [-1,-4,-2,-3,-5] => 2
[1,4,2,5,3] => [1,4,2,5,3] => [-1,-4,-2,-5,-3] => 3
[1,4,3,2,5] => [1,4,3,2,5] => [-1,-4,-3,-2,-5] => 3
[1,4,3,5,2] => [1,4,3,5,2] => [-1,-4,-3,-5,-2] => 4
[1,4,5,2,3] => [1,4,5,2,3] => [-1,-4,-5,-2,-3] => 4
[1,4,5,3,2] => [1,4,5,3,2] => [-1,-4,-5,-3,-2] => 5
[1,5,2,3,4] => [1,5,2,3,4] => [-1,-5,-2,-3,-4] => 3
[1,5,2,4,3] => [1,5,2,4,3] => [-1,-5,-2,-4,-3] => 4
[1,5,3,2,4] => [1,5,3,2,4] => [-1,-5,-3,-2,-4] => 4
[1,5,3,4,2] => [1,5,3,4,2] => [-1,-5,-3,-4,-2] => 5
[1,5,4,2,3] => [1,5,4,2,3] => [-1,-5,-4,-2,-3] => 5
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Description
The number of crossings of a signed permutation.
A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that
  • $i < j \leq \pi(i) < \pi(j)$, or
  • $-i < j \leq -\pi(i) < \pi(j)$, or
  • $i > j > \pi(i) > \pi(j)$.
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Description
Return the signed permutation with all signs reversed.
Map
to signed permutation
Description
The signed permutation with all signs positive.