Identifier
-
Mp00170:
Permutations
—to signed permutation⟶
Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00245: Signed permutations —standardize⟶ Permutations
St001557: Permutations ⟶ ℤ
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] => 1
[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] => 0
[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] => 1
[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] => 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] => [2,1,3,4] => 0
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [3,1,2,4] => 0
[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] => 0
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [3,1,2,4] => 0
[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] => 1
[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] => 1
[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] => 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] => 0
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [4,1,2,3,5] => 0
[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] => 0
[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] => 2
[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] => 1
[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] => 1
[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] => 1
[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] => 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] => 1
[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] => 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] => [1,4,2,3,5] => 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] => 1
[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] => 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] => 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] => 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] => 0
[3,1,4,5,2] => [3,1,4,5,2] => [5,1,3,4,-2] => [4,1,2,3,5] => 0
[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] => 0
[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] => 0
[3,4,1,5,2] => [3,4,1,5,2] => [5,1,2,4,-3] => [4,1,2,3,5] => 0
[3,4,2,1,5] => [3,4,2,1,5] => [3,1,2,5,-4] => [3,1,2,4,5] => 0
[3,4,2,5,1] => [3,4,2,5,1] => [3,1,2,4,-5] => [3,1,2,4,5] => 0
[3,4,5,1,2] => [3,4,5,1,2] => [5,1,2,3,-4] => [4,1,2,3,5] => 0
[3,4,5,2,1] => [3,4,5,2,1] => [4,1,2,3,-5] => [4,1,2,3,5] => 0
[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] => 0
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,5,2,-4] => [3,1,4,2,5] => 0
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Description
The number of inversions of the second entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$
The number of inversions of the first entry is St000054The first entry of the permutation. and the number of inversions of the third entry is St001556The number of inversions of the third entry of a permutation.. The sequence of inversions of all the entries define the Lehmer code of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{2 < k \leq n \mid \pi(2) > \pi(k)\}.$$
The number of inversions of the first entry is St000054The first entry of the permutation. and the number of inversions of the third entry is St001556The number of inversions of the third entry of a permutation.. The sequence of inversions of all the entries define the Lehmer code of a permutation.
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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