Identifier
-
Mp00170:
Permutations
—to signed permutation⟶
Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00245: Signed permutations —standardize⟶ Permutations
St001517: Permutations ⟶ ℤ
Values
[1] => [1] => [-1] => [1] => 0
[1,2] => [1,2] => [2,-1] => [1,2] => 1
[2,1] => [2,1] => [1,-2] => [1,2] => 1
[1,2,3] => [1,2,3] => [2,3,-1] => [1,2,3] => 1
[1,3,2] => [1,3,2] => [3,2,-1] => [2,1,3] => 1
[2,1,3] => [2,1,3] => [1,3,-2] => [1,2,3] => 1
[2,3,1] => [2,3,1] => [1,2,-3] => [1,2,3] => 1
[3,1,2] => [3,1,2] => [3,1,-2] => [2,1,3] => 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] => [1,2,3,4] => 2
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [1,3,2,4] => 2
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [2,1,3,4] => 2
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [3,1,2,4] => 2
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => [2,3,1,4] => 2
[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] => 2
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [1,3,2,4] => 2
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,2,3,4] => 2
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => [1,2,3,4] => 2
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => [1,3,2,4] => 2
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [1,3,2,4] => 2
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => [2,1,3,4] => 2
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [3,1,2,4] => 2
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [2,1,3,4] => 2
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [2,1,3,4] => 2
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => [3,1,2,4] => 2
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [3,1,2,4] => 2
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [2,3,1,4] => 2
[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] => 2
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [2,3,1,4] => 2
[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] => 2
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => [1,2,4,3,5] => 2
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => [1,3,2,4,5] => 2
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => [1,4,2,3,5] => 2
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => [1,3,4,2,5] => 2
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => [1,4,3,2,5] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => [2,1,3,4,5] => 2
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => [2,1,4,3,5] => 2
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => [3,1,2,4,5] => 2
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [4,1,2,3,5] => 2
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => [3,1,4,2,5] => 2
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => [4,1,3,2,5] => 2
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => [2,3,1,4,5] => 2
[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] => 2
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => [4,2,1,3,5] => 2
[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] => 2
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,-1] => [2,4,3,1,5] => 2
[1,5,3,2,4] => [1,5,3,2,4] => [4,3,5,2,-1] => [3,2,4,1,5] => 2
[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] => 2
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => [1,2,4,3,5] => 2
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => [1,3,2,4,5] => 2
[2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => [1,4,2,3,5] => 2
[2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => [1,3,4,2,5] => 2
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => [1,4,3,2,5] => 2
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => [1,2,3,4,5] => 2
[2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => [1,2,4,3,5] => 2
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => [1,2,3,4,5] => 2
[2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => [1,2,3,4,5] => 2
[2,3,5,1,4] => [2,3,5,1,4] => [1,2,5,3,-4] => [1,2,4,3,5] => 2
[2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,-5] => [1,2,4,3,5] => 2
[2,4,1,3,5] => [2,4,1,3,5] => [1,4,2,5,-3] => [1,3,2,4,5] => 2
[2,4,1,5,3] => [2,4,1,5,3] => [1,5,2,4,-3] => [1,4,2,3,5] => 2
[2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,-4] => [1,3,2,4,5] => 2
[2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,-5] => [1,3,2,4,5] => 2
[2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => [1,4,2,3,5] => 2
[2,4,5,3,1] => [2,4,5,3,1] => [1,4,2,3,-5] => [1,4,2,3,5] => 2
[2,5,1,3,4] => [2,5,1,3,4] => [1,4,5,2,-3] => [1,3,4,2,5] => 2
[2,5,1,4,3] => [2,5,1,4,3] => [1,5,4,2,-3] => [1,4,3,2,5] => 2
[2,5,3,1,4] => [2,5,3,1,4] => [1,3,5,2,-4] => [1,3,4,2,5] => 2
[2,5,3,4,1] => [2,5,3,4,1] => [1,3,4,2,-5] => [1,3,4,2,5] => 2
[2,5,4,1,3] => [2,5,4,1,3] => [1,5,3,2,-4] => [1,4,3,2,5] => 2
[2,5,4,3,1] => [2,5,4,3,1] => [1,4,3,2,-5] => [1,4,3,2,5] => 2
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,4,5,-2] => [2,1,3,4,5] => 2
[3,1,2,5,4] => [3,1,2,5,4] => [3,1,5,4,-2] => [2,1,4,3,5] => 2
[3,1,4,2,5] => [3,1,4,2,5] => [4,1,3,5,-2] => [3,1,2,4,5] => 2
[3,1,4,5,2] => [3,1,4,5,2] => [5,1,3,4,-2] => [4,1,2,3,5] => 2
[3,1,5,2,4] => [3,1,5,2,4] => [4,1,5,3,-2] => [3,1,4,2,5] => 2
[3,1,5,4,2] => [3,1,5,4,2] => [5,1,4,3,-2] => [4,1,3,2,5] => 2
[3,2,1,4,5] => [3,2,1,4,5] => [2,1,4,5,-3] => [2,1,3,4,5] => 2
[3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,-3] => [2,1,4,3,5] => 2
[3,2,4,1,5] => [3,2,4,1,5] => [2,1,3,5,-4] => [2,1,3,4,5] => 2
[3,2,4,5,1] => [3,2,4,5,1] => [2,1,3,4,-5] => [2,1,3,4,5] => 2
[3,2,5,1,4] => [3,2,5,1,4] => [2,1,5,3,-4] => [2,1,4,3,5] => 2
[3,2,5,4,1] => [3,2,5,4,1] => [2,1,4,3,-5] => [2,1,4,3,5] => 2
[3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => [3,1,2,4,5] => 2
[3,4,1,5,2] => [3,4,1,5,2] => [5,1,2,4,-3] => [4,1,2,3,5] => 2
[3,4,2,1,5] => [3,4,2,1,5] => [3,1,2,5,-4] => [3,1,2,4,5] => 2
[3,4,2,5,1] => [3,4,2,5,1] => [3,1,2,4,-5] => [3,1,2,4,5] => 2
[3,4,5,1,2] => [3,4,5,1,2] => [5,1,2,3,-4] => [4,1,2,3,5] => 2
[3,4,5,2,1] => [3,4,5,2,1] => [4,1,2,3,-5] => [4,1,2,3,5] => 2
[3,5,1,2,4] => [3,5,1,2,4] => [4,1,5,2,-3] => [3,1,4,2,5] => 2
[3,5,1,4,2] => [3,5,1,4,2] => [5,1,4,2,-3] => [4,1,3,2,5] => 2
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Description
The length of a longest pair of twins in a permutation.
A pair of twins in a permutation is a pair of two disjoint subsequences which are order isomorphic.
A pair of twins in a permutation is a pair of two disjoint subsequences which are order isomorphic.
Map
to signed permutation
Description
The signed permutation with all signs positive.
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 $\pi(1),\dots,\pi(n)$.
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 $2n$.
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 $2n$.
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