Identifier
-
Mp00089:
Permutations
—Inverse Kreweras complement⟶
Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00168: Signed permutations —Kreweras complement⟶ Signed permutations
St001427: Signed permutations ⟶ ℤ
Values
[1] => [1] => [1] => [-1] => 1
[1,2] => [2,1] => [2,1] => [-1,2] => 1
[2,1] => [1,2] => [1,2] => [2,-1] => 1
[1,2,3] => [2,3,1] => [2,3,1] => [-1,2,3] => 1
[1,3,2] => [3,2,1] => [3,2,1] => [-1,3,2] => 2
[2,1,3] => [1,3,2] => [1,3,2] => [2,-1,3] => 1
[2,3,1] => [1,2,3] => [1,2,3] => [2,3,-1] => 1
[3,1,2] => [3,1,2] => [3,1,2] => [3,-1,2] => 1
[3,2,1] => [2,1,3] => [2,1,3] => [3,2,-1] => 2
[1,2,3,4] => [2,3,4,1] => [2,3,4,1] => [-1,2,3,4] => 1
[1,2,4,3] => [2,4,3,1] => [2,4,3,1] => [-1,2,4,3] => 2
[1,3,2,4] => [3,2,4,1] => [3,2,4,1] => [-1,3,2,4] => 2
[1,3,4,2] => [4,2,3,1] => [4,2,3,1] => [-1,3,4,2] => 2
[1,4,2,3] => [3,4,2,1] => [3,4,2,1] => [-1,4,2,3] => 2
[1,4,3,2] => [4,3,2,1] => [4,3,2,1] => [-1,4,3,2] => 3
[2,1,3,4] => [1,3,4,2] => [1,3,4,2] => [2,-1,3,4] => 1
[2,1,4,3] => [1,4,3,2] => [1,4,3,2] => [2,-1,4,3] => 2
[2,3,1,4] => [1,2,4,3] => [1,2,4,3] => [2,3,-1,4] => 1
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 1
[2,4,1,3] => [1,4,2,3] => [1,4,2,3] => [2,4,-1,3] => 1
[2,4,3,1] => [1,3,2,4] => [1,3,2,4] => [2,4,3,-1] => 2
[3,1,2,4] => [3,1,4,2] => [3,1,4,2] => [3,-1,2,4] => 1
[3,1,4,2] => [4,1,3,2] => [4,1,3,2] => [3,-1,4,2] => 2
[3,2,1,4] => [2,1,4,3] => [2,1,4,3] => [3,2,-1,4] => 2
[3,2,4,1] => [2,1,3,4] => [2,1,3,4] => [3,2,4,-1] => 2
[3,4,1,2] => [4,1,2,3] => [4,1,2,3] => [3,4,-1,2] => 1
[3,4,2,1] => [3,1,2,4] => [3,1,2,4] => [3,4,2,-1] => 2
[4,1,2,3] => [3,4,1,2] => [3,4,1,2] => [4,-1,2,3] => 1
[4,1,3,2] => [4,3,1,2] => [4,3,1,2] => [4,-1,3,2] => 2
[4,2,1,3] => [2,4,1,3] => [2,4,1,3] => [4,2,-1,3] => 2
[4,2,3,1] => [2,3,1,4] => [2,3,1,4] => [4,2,3,-1] => 2
[4,3,1,2] => [4,2,1,3] => [4,2,1,3] => [4,3,-1,2] => 2
[4,3,2,1] => [3,2,1,4] => [3,2,1,4] => [4,3,2,-1] => 3
[1,2,3,4,5] => [2,3,4,5,1] => [2,3,4,5,1] => [-1,2,3,4,5] => 1
[1,2,3,5,4] => [2,3,5,4,1] => [2,3,5,4,1] => [-1,2,3,5,4] => 2
[1,2,4,3,5] => [2,4,3,5,1] => [2,4,3,5,1] => [-1,2,4,3,5] => 2
[1,2,4,5,3] => [2,5,3,4,1] => [2,5,3,4,1] => [-1,2,4,5,3] => 2
[1,2,5,3,4] => [2,4,5,3,1] => [2,4,5,3,1] => [-1,2,5,3,4] => 2
[1,2,5,4,3] => [2,5,4,3,1] => [2,5,4,3,1] => [-1,2,5,4,3] => 3
[1,3,2,4,5] => [3,2,4,5,1] => [3,2,4,5,1] => [-1,3,2,4,5] => 2
[1,3,2,5,4] => [3,2,5,4,1] => [3,2,5,4,1] => [-1,3,2,5,4] => 3
[1,3,4,2,5] => [4,2,3,5,1] => [4,2,3,5,1] => [-1,3,4,2,5] => 2
[1,3,4,5,2] => [5,2,3,4,1] => [5,2,3,4,1] => [-1,3,4,5,2] => 2
[1,3,5,2,4] => [4,2,5,3,1] => [4,2,5,3,1] => [-1,3,5,2,4] => 2
[1,3,5,4,2] => [5,2,4,3,1] => [5,2,4,3,1] => [-1,3,5,4,2] => 3
[1,4,2,3,5] => [3,4,2,5,1] => [3,4,2,5,1] => [-1,4,2,3,5] => 2
[1,4,2,5,3] => [3,5,2,4,1] => [3,5,2,4,1] => [-1,4,2,5,3] => 3
[1,4,3,2,5] => [4,3,2,5,1] => [4,3,2,5,1] => [-1,4,3,2,5] => 3
[1,4,3,5,2] => [5,3,2,4,1] => [5,3,2,4,1] => [-1,4,3,5,2] => 3
[1,4,5,2,3] => [4,5,2,3,1] => [4,5,2,3,1] => [-1,4,5,2,3] => 2
[1,4,5,3,2] => [5,4,2,3,1] => [5,4,2,3,1] => [-1,4,5,3,2] => 3
[1,5,2,3,4] => [3,4,5,2,1] => [3,4,5,2,1] => [-1,5,2,3,4] => 2
[1,5,2,4,3] => [3,5,4,2,1] => [3,5,4,2,1] => [-1,5,2,4,3] => 3
[1,5,3,2,4] => [4,3,5,2,1] => [4,3,5,2,1] => [-1,5,3,2,4] => 3
[1,5,3,4,2] => [5,3,4,2,1] => [5,3,4,2,1] => [-1,5,3,4,2] => 3
[1,5,4,2,3] => [4,5,3,2,1] => [4,5,3,2,1] => [-1,5,4,2,3] => 3
[1,5,4,3,2] => [5,4,3,2,1] => [5,4,3,2,1] => [-1,5,4,3,2] => 4
[2,1,3,4,5] => [1,3,4,5,2] => [1,3,4,5,2] => [2,-1,3,4,5] => 1
[2,1,3,5,4] => [1,3,5,4,2] => [1,3,5,4,2] => [2,-1,3,5,4] => 2
[2,1,4,3,5] => [1,4,3,5,2] => [1,4,3,5,2] => [2,-1,4,3,5] => 2
[2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => [2,-1,4,5,3] => 2
[2,1,5,3,4] => [1,4,5,3,2] => [1,4,5,3,2] => [2,-1,5,3,4] => 2
[2,1,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => [2,-1,5,4,3] => 3
[2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => [2,3,-1,4,5] => 1
[2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => [2,3,-1,5,4] => 2
[2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => [2,3,4,-1,5] => 1
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => 1
[2,3,5,1,4] => [1,2,5,3,4] => [1,2,5,3,4] => [2,3,5,-1,4] => 1
[2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => [2,3,5,4,-1] => 2
[2,4,1,3,5] => [1,4,2,5,3] => [1,4,2,5,3] => [2,4,-1,3,5] => 1
[2,4,1,5,3] => [1,5,2,4,3] => [1,5,2,4,3] => [2,4,-1,5,3] => 2
[2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => [2,4,3,-1,5] => 2
[2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => [2,4,3,5,-1] => 2
[2,4,5,1,3] => [1,5,2,3,4] => [1,5,2,3,4] => [2,4,5,-1,3] => 1
[2,4,5,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => [2,4,5,3,-1] => 2
[2,5,1,3,4] => [1,4,5,2,3] => [1,4,5,2,3] => [2,5,-1,3,4] => 1
[2,5,1,4,3] => [1,5,4,2,3] => [1,5,4,2,3] => [2,5,-1,4,3] => 2
[2,5,3,1,4] => [1,3,5,2,4] => [1,3,5,2,4] => [2,5,3,-1,4] => 2
[2,5,3,4,1] => [1,3,4,2,5] => [1,3,4,2,5] => [2,5,3,4,-1] => 2
[2,5,4,1,3] => [1,5,3,2,4] => [1,5,3,2,4] => [2,5,4,-1,3] => 2
[2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => [2,5,4,3,-1] => 3
[3,1,2,4,5] => [3,1,4,5,2] => [3,1,4,5,2] => [3,-1,2,4,5] => 1
[3,1,2,5,4] => [3,1,5,4,2] => [3,1,5,4,2] => [3,-1,2,5,4] => 2
[3,1,4,2,5] => [4,1,3,5,2] => [4,1,3,5,2] => [3,-1,4,2,5] => 2
[3,1,4,5,2] => [5,1,3,4,2] => [5,1,3,4,2] => [3,-1,4,5,2] => 2
[3,1,5,2,4] => [4,1,5,3,2] => [4,1,5,3,2] => [3,-1,5,2,4] => 2
[3,1,5,4,2] => [5,1,4,3,2] => [5,1,4,3,2] => [3,-1,5,4,2] => 3
[3,2,1,4,5] => [2,1,4,5,3] => [2,1,4,5,3] => [3,2,-1,4,5] => 2
[3,2,1,5,4] => [2,1,5,4,3] => [2,1,5,4,3] => [3,2,-1,5,4] => 3
[3,2,4,1,5] => [2,1,3,5,4] => [2,1,3,5,4] => [3,2,4,-1,5] => 2
[3,2,4,5,1] => [2,1,3,4,5] => [2,1,3,4,5] => [3,2,4,5,-1] => 2
[3,2,5,1,4] => [2,1,5,3,4] => [2,1,5,3,4] => [3,2,5,-1,4] => 2
[3,2,5,4,1] => [2,1,4,3,5] => [2,1,4,3,5] => [3,2,5,4,-1] => 3
[3,4,1,2,5] => [4,1,2,5,3] => [4,1,2,5,3] => [3,4,-1,2,5] => 1
[3,4,1,5,2] => [5,1,2,4,3] => [5,1,2,4,3] => [3,4,-1,5,2] => 2
[3,4,2,1,5] => [3,1,2,5,4] => [3,1,2,5,4] => [3,4,2,-1,5] => 2
[3,4,2,5,1] => [3,1,2,4,5] => [3,1,2,4,5] => [3,4,2,5,-1] => 2
[3,4,5,1,2] => [5,1,2,3,4] => [5,1,2,3,4] => [3,4,5,-1,2] => 1
[3,4,5,2,1] => [4,1,2,3,5] => [4,1,2,3,5] => [3,4,5,2,-1] => 2
[3,5,1,2,4] => [4,1,5,2,3] => [4,1,5,2,3] => [3,5,-1,2,4] => 1
[3,5,1,4,2] => [5,1,4,2,3] => [5,1,4,2,3] => [3,5,-1,4,2] => 2
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Description
The number of descents of a signed permutation.
A descent of a signed permutation $\sigma$ of length $n$ is an index $0 \leq i < n$ such that $\sigma(i) > \sigma(i+1)$, setting $\sigma(0) = 0$.
A descent of a signed permutation $\sigma$ of length $n$ is an index $0 \leq i < n$ such that $\sigma(i) > \sigma(i+1)$, setting $\sigma(0) = 0$.
Map
Inverse Kreweras complement
Description
Sends the permutation $\pi \in \mathfrak{S}_n$ to the permutation $c\pi^{-1}$ where $c = (1,\ldots,n)$ is the long cycle.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
Kreweras complement
Description
The Kreweras complement of a signed permutation.
This is the signed permutation $\pi^{-1}c$ where $c = (1,\ldots,n,-1,-2,\dots,-n)$ is the long cycle.
The order of the Kreweras complement on signed permutations of $\{\pm 1,\dots, \pm n\}$ is $2n$.
This is the signed permutation $\pi^{-1}c$ where $c = (1,\ldots,n,-1,-2,\dots,-n)$ is the long cycle.
The order of the Kreweras complement on signed permutations of $\{\pm 1,\dots, \pm n\}$ is $2n$.
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