Identifier
-
Mp00277:
Permutations
—catalanization⟶
Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001769: Signed permutations ⟶ ℤ
Values
[1] => [1] => [1] => [1] => 0
[1,2] => [1,2] => [2,1] => [2,1] => 1
[2,1] => [2,1] => [1,2] => [1,2] => 0
[1,2,3] => [1,2,3] => [2,3,1] => [2,3,1] => 2
[1,3,2] => [1,3,2] => [3,2,1] => [3,2,1] => 1
[2,1,3] => [2,1,3] => [1,3,2] => [1,3,2] => 1
[2,3,1] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[3,1,2] => [2,3,1] => [1,2,3] => [1,2,3] => 0
[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] => [2,3,4,1] => 3
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => [2,4,3,1] => 2
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => [3,2,4,1] => 2
[1,3,4,2] => [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 1
[1,4,2,3] => [1,3,4,2] => [4,2,3,1] => [4,2,3,1] => 1
[1,4,3,2] => [1,4,3,2] => [4,3,2,1] => [4,3,2,1] => 2
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => [1,3,4,2] => 2
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => [1,4,3,2] => 1
[2,3,1,4] => [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 1
[2,3,4,1] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[2,4,1,3] => [4,3,1,2] => [4,2,1,3] => [4,2,1,3] => 2
[2,4,3,1] => [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 1
[3,1,2,4] => [2,3,1,4] => [1,2,4,3] => [1,2,4,3] => 1
[3,1,4,2] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[3,2,1,4] => [3,2,1,4] => [2,1,4,3] => [2,1,4,3] => 2
[3,2,4,1] => [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 1
[3,4,1,2] => [4,3,2,1] => [3,2,1,4] => [3,2,1,4] => 1
[3,4,2,1] => [3,4,2,1] => [3,1,2,4] => [3,1,2,4] => 2
[4,1,2,3] => [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[4,1,3,2] => [2,4,3,1] => [1,3,2,4] => [1,3,2,4] => 1
[4,2,1,3] => [3,2,4,1] => [2,1,3,4] => [2,1,3,4] => 1
[4,2,3,1] => [3,4,2,1] => [3,1,2,4] => [3,1,2,4] => 2
[4,3,1,2] => [3,4,2,1] => [3,1,2,4] => [3,1,2,4] => 2
[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] => [1,3,4,5,2] => 3
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,2] => [1,3,5,4,2] => 2
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,2] => [1,4,3,5,2] => 2
[2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[2,1,5,3,4] => [2,1,4,5,3] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => 2
[2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[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] => [2,5,4,1,3] => [1,5,3,2,4] => [1,5,3,2,4] => 2
[2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[2,4,5,3,1] => [2,4,5,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[2,5,3,4,1] => [2,4,5,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[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] => [2,3,1,4,5] => [1,2,4,5,3] => [1,2,4,5,3] => 2
[3,1,2,5,4] => [2,3,1,5,4] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[3,1,4,2,5] => [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[3,1,4,5,2] => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[3,1,5,2,4] => [2,5,4,1,3] => [1,5,3,2,4] => [1,5,3,2,4] => 2
[3,1,5,4,2] => [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[4,1,2,3,5] => [2,3,4,1,5] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[4,1,2,5,3] => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1,3,2,5] => [2,4,3,1,5] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[4,1,3,5,2] => [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[4,1,5,2,3] => [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[4,1,5,3,2] => [2,4,5,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[5,1,2,3,4] => [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[5,1,2,4,3] => [2,3,5,4,1] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[5,1,3,2,4] => [2,4,3,5,1] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[5,1,3,4,2] => [2,4,5,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[5,1,4,2,3] => [2,4,5,3,1] => [1,4,2,3,5] => [1,4,2,3,5] => 2
[5,1,4,3,2] => [2,5,4,3,1] => [1,4,3,2,5] => [1,4,3,2,5] => 1
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Description
The reflection length of a signed permutation.
This is the minimal numbers of reflections needed to express a signed permutation.
This is the minimal numbers of reflections needed to express a signed permutation.
Map
to signed permutation
Description
The signed permutation with all signs positive.
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
catalanization
Description
The catalanization of a permutation.
For a permutation $\sigma$, this is the product of the reflections corresponding to the inversions of $\sigma$ in lex-order.
A permutation is $231$-avoiding if and only if it is a fixpoint of this map. Also, for every permutation there exists an index $k$ such that the $k$-fold application of this map is $231$-avoiding.
For a permutation $\sigma$, this is the product of the reflections corresponding to the inversions of $\sigma$ in lex-order.
A permutation is $231$-avoiding if and only if it is a fixpoint of this map. Also, for every permutation there exists an index $k$ such that the $k$-fold application of this map is $231$-avoiding.
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