Identifier
Values
[1] => [1] => [-1] => [-1] => 0
[1,2] => [1,2] => [2,-1] => [-2,1] => 0
[2,1] => [2,1] => [1,-2] => [1,-2] => 0
[1,2,3] => [1,2,3] => [2,3,-1] => [-3,1,2] => 0
[1,3,2] => [1,3,2] => [3,2,-1] => [-3,2,1] => 0
[2,1,3] => [2,1,3] => [1,3,-2] => [1,-3,2] => 0
[2,3,1] => [2,3,1] => [1,2,-3] => [1,2,-3] => 0
[3,1,2] => [3,1,2] => [3,1,-2] => [2,-3,1] => 1
[3,2,1] => [3,2,1] => [2,1,-3] => [2,1,-3] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => [-4,1,2,3] => 0
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [-4,1,3,2] => 0
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [-4,2,1,3] => 0
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [-4,2,3,1] => 0
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => [-4,3,1,2] => 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => [-4,3,2,1] => 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => [1,-4,2,3] => 0
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [1,-4,3,2] => 0
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,2,-4,3] => 0
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => [1,2,3,-4] => 0
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => [1,3,-4,2] => 1
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [1,3,2,-4] => 1
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => [2,-4,1,3] => 1
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [2,-4,3,1] => 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [2,1,-4,3] => 1
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [2,1,3,-4] => 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => [2,3,-4,1] => 2
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [2,3,1,-4] => 2
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [3,-4,1,2] => 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => [3,-4,2,1] => 1
[4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => [3,1,-4,2] => 1
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [3,1,2,-4] => 1
[4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => [3,2,-4,1] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => [3,2,1,-4] => 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => [1,-5,2,3,4] => 0
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => [1,-5,2,4,3] => 0
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => [1,-5,3,2,4] => 0
[2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => [1,-5,3,4,2] => 0
[2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => [1,-5,4,2,3] => 1
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => [1,-5,4,3,2] => 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => [1,2,-5,3,4] => 0
[2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => [1,2,-5,4,3] => 0
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => [1,2,3,-5,4] => 0
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => [1,2,3,4,-5] => 0
[2,3,5,1,4] => [2,3,5,1,4] => [1,2,5,3,-4] => [1,2,4,-5,3] => 1
[2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,-5] => [1,2,4,3,-5] => 1
[2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,-3] => [1,3,-5,2,4] => 1
[2,4,1,5,3] => [2,4,1,5,3] => [1,5,2,4,-3] => [1,3,-5,4,2] => 1
[2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,-4] => [1,3,2,-5,4] => 1
[2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,-5] => [1,3,2,4,-5] => 1
[2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => [1,3,4,-5,2] => 2
[2,4,5,3,1] => [2,4,5,3,1] => [1,4,2,3,-5] => [1,3,4,2,-5] => 2
[2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,-3] => [1,4,-5,2,3] => 1
[2,5,1,4,3] => [2,5,1,4,3] => [1,5,4,2,-3] => [1,4,-5,3,2] => 1
[2,5,3,1,4] => [2,5,3,1,4] => [1,3,5,2,-4] => [1,4,2,-5,3] => 1
[2,5,3,4,1] => [2,5,3,4,1] => [1,3,4,2,-5] => [1,4,2,3,-5] => 1
[2,5,4,1,3] => [2,5,4,1,3] => [1,5,3,2,-4] => [1,4,3,-5,2] => 1
[2,5,4,3,1] => [2,5,4,3,1] => [1,4,3,2,-5] => [1,4,3,2,-5] => 1
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Description
The number of excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
Map
inverse Kreweras complement
Description
The inverse Kreweras complement of a signed permutation.
This is the signed permutation $c \pi^{-1}$ where $c = (1,\ldots,n,-1,-2,\dots,-n)$ is the long cycle.
The order of the inverse Kreweras complement on signed permutations of $\{\pm 1,\dots, \pm n\}$ is $2n$.
Map
inverse
Description
The inverse of a signed permutation.
Map
to signed permutation
Description
The signed permutation with all signs positive.