Identifier
Values
[1,2] => [1,2] => [2,-1] => [1,2] => 0
[2,1] => [2,1] => [1,-2] => [1,2] => 0
[1,2,3] => [1,2,3] => [2,3,-1] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [3,2,-1] => [2,1,3] => 0
[2,1,3] => [2,1,3] => [1,3,-2] => [1,2,3] => 0
[2,3,1] => [2,3,1] => [1,2,-3] => [1,2,3] => 0
[3,1,2] => [3,1,2] => [3,1,-2] => [2,1,3] => 0
[3,2,1] => [3,2,1] => [2,1,-3] => [2,1,3] => 0
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [1,3,2,4] => 0
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [2,1,3,4] => 0
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [3,1,2,4] => 1
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => [2,3,1,4] => 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => [3,2,1,4] => 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => [1,2,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [1,3,2,4] => 0
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,2,3,4] => 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] => [1,3,2,4] => 0
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [1,3,2,4] => 0
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => [2,1,3,4] => 0
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [3,1,2,4] => 1
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [2,1,3,4] => 0
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [2,1,3,4] => 0
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => [3,1,2,4] => 1
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [3,1,2,4] => 1
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [2,3,1,4] => 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => [3,2,1,4] => 1
[4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => [2,3,1,4] => 1
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [2,3,1,4] => 1
[4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => [3,2,1,4] => 1
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => [3,2,1,4] => 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => [1,2,4,3,5] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => [1,3,2,4,5] => 0
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => [1,4,2,3,5] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => [1,3,4,2,5] => 1
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => [1,4,3,2,5] => 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => [2,1,3,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => [2,1,4,3,5] => 0
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => [3,1,2,4,5] => 1
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [4,1,2,3,5] => 1
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => [3,1,4,2,5] => 0
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => [4,1,3,2,5] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => [2,3,1,4,5] => 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => [2,4,1,3,5] => 2
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,-1] => [3,2,1,4,5] => 1
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => [4,2,1,3,5] => 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => [3,4,1,2,5] => 2
[1,4,5,3,2] => [1,4,5,3,2] => [5,4,2,3,-1] => [4,3,1,2,5] => 2
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,-1] => [2,3,4,1,5] => 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,-1] => [2,4,3,1,5] => 1
[1,5,3,2,4] => [1,5,3,2,4] => [4,3,5,2,-1] => [3,2,4,1,5] => 1
[1,5,3,4,2] => [1,5,3,4,2] => [5,3,4,2,-1] => [4,2,3,1,5] => 2
[1,5,4,2,3] => [1,5,4,2,3] => [4,5,3,2,-1] => [3,4,2,1,5] => 2
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,-1] => [4,3,2,1,5] => 2
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => [1,2,3,4,5] => 0
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => [1,2,4,3,5] => 0
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => [1,3,2,4,5] => 0
[2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => [1,4,2,3,5] => 1
[2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => [1,3,4,2,5] => 1
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => [1,4,3,2,5] => 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => [1,2,3,4,5] => 0
[2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => [1,2,4,3,5] => 0
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => [1,2,3,4,5] => 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] => [1,2,4,3,5] => 0
[2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,-5] => [1,2,4,3,5] => 0
[2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,-3] => [1,3,2,4,5] => 0
[2,4,1,5,3] => [2,4,1,5,3] => [1,5,2,4,-3] => [1,4,2,3,5] => 1
[2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,-4] => [1,3,2,4,5] => 0
[2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,-5] => [1,3,2,4,5] => 0
[2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => [1,4,2,3,5] => 1
[2,4,5,3,1] => [2,4,5,3,1] => [1,4,2,3,-5] => [1,4,2,3,5] => 1
[2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,-3] => [1,3,4,2,5] => 1
[2,5,1,4,3] => [2,5,1,4,3] => [1,5,4,2,-3] => [1,4,3,2,5] => 1
[2,5,3,1,4] => [2,5,3,1,4] => [1,3,5,2,-4] => [1,3,4,2,5] => 1
[2,5,3,4,1] => [2,5,3,4,1] => [1,3,4,2,-5] => [1,3,4,2,5] => 1
[2,5,4,1,3] => [2,5,4,1,3] => [1,5,3,2,-4] => [1,4,3,2,5] => 1
[2,5,4,3,1] => [2,5,4,3,1] => [1,4,3,2,-5] => [1,4,3,2,5] => 1
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,4,5,-2] => [2,1,3,4,5] => 0
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,5,4,-2] => [2,1,4,3,5] => 0
[3,1,4,2,5] => [3,1,4,2,5] => [4,1,3,5,-2] => [3,1,2,4,5] => 1
[3,1,4,5,2] => [3,1,4,5,2] => [5,1,3,4,-2] => [4,1,2,3,5] => 1
[3,1,5,2,4] => [3,1,5,2,4] => [4,1,5,3,-2] => [3,1,4,2,5] => 0
[3,1,5,4,2] => [3,1,5,4,2] => [5,1,4,3,-2] => [4,1,3,2,5] => 1
[3,2,1,4,5] => [3,2,1,4,5] => [2,1,4,5,-3] => [2,1,3,4,5] => 0
[3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,-3] => [2,1,4,3,5] => 0
[3,2,4,1,5] => [3,2,4,1,5] => [2,1,3,5,-4] => [2,1,3,4,5] => 0
[3,2,4,5,1] => [3,2,4,5,1] => [2,1,3,4,-5] => [2,1,3,4,5] => 0
[3,2,5,1,4] => [3,2,5,1,4] => [2,1,5,3,-4] => [2,1,4,3,5] => 0
[3,2,5,4,1] => [3,2,5,4,1] => [2,1,4,3,-5] => [2,1,4,3,5] => 0
[3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => [3,1,2,4,5] => 1
[3,4,1,5,2] => [3,4,1,5,2] => [5,1,2,4,-3] => [4,1,2,3,5] => 1
[3,4,2,1,5] => [3,4,2,1,5] => [3,1,2,5,-4] => [3,1,2,4,5] => 1
[3,4,2,5,1] => [3,4,2,5,1] => [3,1,2,4,-5] => [3,1,2,4,5] => 1
[3,4,5,1,2] => [3,4,5,1,2] => [5,1,2,3,-4] => [4,1,2,3,5] => 1
[3,4,5,2,1] => [3,4,5,2,1] => [4,1,2,3,-5] => [4,1,2,3,5] => 1
[3,5,1,2,4] => [3,5,1,2,4] => [4,1,5,2,-3] => [3,1,4,2,5] => 0
[3,5,1,4,2] => [3,5,1,4,2] => [5,1,4,2,-3] => [4,1,3,2,5] => 1
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,5,2,-4] => [3,1,4,2,5] => 0
>>> Load all 152 entries. <<<
[3,5,2,4,1] => [3,5,2,4,1] => [3,1,4,2,-5] => [3,1,4,2,5] => 0
[3,5,4,1,2] => [3,5,4,1,2] => [5,1,3,2,-4] => [4,1,3,2,5] => 1
[3,5,4,2,1] => [3,5,4,2,1] => [4,1,3,2,-5] => [4,1,3,2,5] => 1
[4,1,2,3,5] => [4,1,2,3,5] => [3,4,1,5,-2] => [2,3,1,4,5] => 1
[4,1,2,5,3] => [4,1,2,5,3] => [3,5,1,4,-2] => [2,4,1,3,5] => 2
[4,1,3,2,5] => [4,1,3,2,5] => [4,3,1,5,-2] => [3,2,1,4,5] => 1
[4,1,3,5,2] => [4,1,3,5,2] => [5,3,1,4,-2] => [4,2,1,3,5] => 1
[4,1,5,2,3] => [4,1,5,2,3] => [4,5,1,3,-2] => [3,4,1,2,5] => 2
[4,1,5,3,2] => [4,1,5,3,2] => [5,4,1,3,-2] => [4,3,1,2,5] => 2
[4,2,1,3,5] => [4,2,1,3,5] => [2,4,1,5,-3] => [2,3,1,4,5] => 1
[4,2,1,5,3] => [4,2,1,5,3] => [2,5,1,4,-3] => [2,4,1,3,5] => 2
[4,2,3,1,5] => [4,2,3,1,5] => [2,3,1,5,-4] => [2,3,1,4,5] => 1
[4,2,3,5,1] => [4,2,3,5,1] => [2,3,1,4,-5] => [2,3,1,4,5] => 1
[4,2,5,1,3] => [4,2,5,1,3] => [2,5,1,3,-4] => [2,4,1,3,5] => 2
[4,2,5,3,1] => [4,2,5,3,1] => [2,4,1,3,-5] => [2,4,1,3,5] => 2
[4,3,1,2,5] => [4,3,1,2,5] => [4,2,1,5,-3] => [3,2,1,4,5] => 1
[4,3,1,5,2] => [4,3,1,5,2] => [5,2,1,4,-3] => [4,2,1,3,5] => 1
[4,3,2,1,5] => [4,3,2,1,5] => [3,2,1,5,-4] => [3,2,1,4,5] => 1
[4,3,2,5,1] => [4,3,2,5,1] => [3,2,1,4,-5] => [3,2,1,4,5] => 1
[4,3,5,1,2] => [4,3,5,1,2] => [5,2,1,3,-4] => [4,2,1,3,5] => 1
[4,3,5,2,1] => [4,3,5,2,1] => [4,2,1,3,-5] => [4,2,1,3,5] => 1
[4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => [3,4,1,2,5] => 2
[4,5,1,3,2] => [4,5,1,3,2] => [5,4,1,2,-3] => [4,3,1,2,5] => 2
[4,5,2,1,3] => [4,5,2,1,3] => [3,5,1,2,-4] => [3,4,1,2,5] => 2
[4,5,2,3,1] => [4,5,2,3,1] => [3,4,1,2,-5] => [3,4,1,2,5] => 2
[4,5,3,1,2] => [4,5,3,1,2] => [5,3,1,2,-4] => [4,3,1,2,5] => 2
[4,5,3,2,1] => [4,5,3,2,1] => [4,3,1,2,-5] => [4,3,1,2,5] => 2
[5,1,2,3,4] => [5,1,2,3,4] => [3,4,5,1,-2] => [2,3,4,1,5] => 1
[5,1,2,4,3] => [5,1,2,4,3] => [3,5,4,1,-2] => [2,4,3,1,5] => 1
[5,1,3,2,4] => [5,1,3,2,4] => [4,3,5,1,-2] => [3,2,4,1,5] => 1
[5,1,3,4,2] => [5,1,3,4,2] => [5,3,4,1,-2] => [4,2,3,1,5] => 2
[5,1,4,2,3] => [5,1,4,2,3] => [4,5,3,1,-2] => [3,4,2,1,5] => 2
[5,1,4,3,2] => [5,1,4,3,2] => [5,4,3,1,-2] => [4,3,2,1,5] => 2
[5,2,1,3,4] => [5,2,1,3,4] => [2,4,5,1,-3] => [2,3,4,1,5] => 1
[5,2,1,4,3] => [5,2,1,4,3] => [2,5,4,1,-3] => [2,4,3,1,5] => 1
[5,2,3,1,4] => [5,2,3,1,4] => [2,3,5,1,-4] => [2,3,4,1,5] => 1
[5,2,3,4,1] => [5,2,3,4,1] => [2,3,4,1,-5] => [2,3,4,1,5] => 1
[5,2,4,1,3] => [5,2,4,1,3] => [2,5,3,1,-4] => [2,4,3,1,5] => 1
[5,2,4,3,1] => [5,2,4,3,1] => [2,4,3,1,-5] => [2,4,3,1,5] => 1
[5,3,1,2,4] => [5,3,1,2,4] => [4,2,5,1,-3] => [3,2,4,1,5] => 1
[5,3,1,4,2] => [5,3,1,4,2] => [5,2,4,1,-3] => [4,2,3,1,5] => 2
[5,3,2,1,4] => [5,3,2,1,4] => [3,2,5,1,-4] => [3,2,4,1,5] => 1
[5,3,2,4,1] => [5,3,2,4,1] => [3,2,4,1,-5] => [3,2,4,1,5] => 1
[5,3,4,1,2] => [5,3,4,1,2] => [5,2,3,1,-4] => [4,2,3,1,5] => 2
[5,3,4,2,1] => [5,3,4,2,1] => [4,2,3,1,-5] => [4,2,3,1,5] => 2
[5,4,1,2,3] => [5,4,1,2,3] => [4,5,2,1,-3] => [3,4,2,1,5] => 2
[5,4,1,3,2] => [5,4,1,3,2] => [5,4,2,1,-3] => [4,3,2,1,5] => 2
[5,4,2,1,3] => [5,4,2,1,3] => [3,5,2,1,-4] => [3,4,2,1,5] => 2
[5,4,2,3,1] => [5,4,2,3,1] => [3,4,2,1,-5] => [3,4,2,1,5] => 2
[5,4,3,1,2] => [5,4,3,1,2] => [5,3,2,1,-4] => [4,3,2,1,5] => 2
[5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1,-5] => [4,3,2,1,5] => 2
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Description
The number of even inversions of a permutation.
An inversion $i < j$ of a permutation is even if $i \equiv j~(\operatorname{mod} 2)$. See St000539The number of odd inversions of a permutation. for odd inversions.
Map
standardize
Description
Return the standardization of the signed permutation, where 1 is the smallest and -1 the largest element.
Let $\pi\in\mathfrak H_n$ be a signed permutation. Assuming the order $1 < \dots < n < -n < \dots < -1$, this map returns the permutation in $\mathfrak S_n$ which is order isomorphic to $\pi(1),\dots,\pi(n)$.
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
to signed permutation
Description
The signed permutation with all signs positive.