- FindStat

Identifier

Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St001087: Permutations ⟶ ℤ

Values

[1] => [1] => [1] => [1] => 0

[1,2] => [1,2] => [1,2] => [2,1] => 0

[2,1] => [2,1] => [2,1] => [1,2] => 0

[1,2,3] => [1,2,3] => [1,3,2] => [3,2,1] => 0

[1,3,2] => [1,3,2] => [1,3,2] => [3,2,1] => 0

[2,1,3] => [2,1,3] => [2,1,3] => [1,3,2] => 0

[2,3,1] => [1,3,2] => [1,3,2] => [3,2,1] => 0

[3,1,2] => [3,1,2] => [3,1,2] => [3,1,2] => 0

[3,2,1] => [3,2,1] => [3,2,1] => [2,1,3] => 0

[1,2,3,4] => [1,2,3,4] => [1,4,3,2] => [4,3,2,1] => 0

[1,2,4,3] => [1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 0

[1,3,2,4] => [1,3,2,4] => [1,4,3,2] => [4,3,2,1] => 0

[1,3,4,2] => [1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 0

[1,4,2,3] => [1,4,2,3] => [1,4,3,2] => [4,3,2,1] => 0

[1,4,3,2] => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 0

[2,1,3,4] => [2,1,3,4] => [2,1,4,3] => [1,4,3,2] => 0

[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 0

[2,3,1,4] => [1,3,2,4] => [1,4,3,2] => [4,3,2,1] => 0

[2,3,4,1] => [1,2,4,3] => [1,4,3,2] => [4,3,2,1] => 0

[2,4,1,3] => [2,4,1,3] => [2,4,1,3] => [1,4,2,3] => 1

[2,4,3,1] => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 0

[3,1,2,4] => [3,1,2,4] => [3,1,4,2] => [4,1,3,2] => 0

[3,1,4,2] => [2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 0

[3,2,1,4] => [3,2,1,4] => [3,2,1,4] => [2,1,4,3] => 0

[3,2,4,1] => [2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 0

[3,4,1,2] => [2,4,1,3] => [2,4,1,3] => [1,4,2,3] => 1

[3,4,2,1] => [1,4,3,2] => [1,4,3,2] => [4,3,2,1] => 0

[4,1,2,3] => [4,1,2,3] => [4,1,3,2] => [4,3,1,2] => 0

[4,1,3,2] => [4,1,3,2] => [4,1,3,2] => [4,3,1,2] => 0

[4,2,1,3] => [4,2,1,3] => [4,2,1,3] => [2,4,1,3] => 0

[4,2,3,1] => [4,1,3,2] => [4,1,3,2] => [4,3,1,2] => 0

[4,3,1,2] => [4,3,1,2] => [4,3,1,2] => [4,2,1,3] => 0

[4,3,2,1] => [4,3,2,1] => [4,3,2,1] => [3,2,1,4] => 0

[1,2,3,4,5] => [1,2,3,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,2,3,5,4] => [1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,2,4,3,5] => [1,2,4,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,2,4,5,3] => [1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,2,5,3,4] => [1,2,5,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,2,5,4,3] => [1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,3,2,4,5] => [1,3,2,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,3,2,5,4] => [1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,3,4,2,5] => [1,2,4,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,3,4,5,2] => [1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,3,5,2,4] => [1,3,5,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,3,5,4,2] => [1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,4,2,3,5] => [1,4,2,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,4,2,5,3] => [1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,4,3,2,5] => [1,4,3,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,4,3,5,2] => [1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,4,5,2,3] => [1,3,5,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,4,5,3,2] => [1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,5,2,3,4] => [1,5,2,3,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,5,2,4,3] => [1,5,2,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,5,3,2,4] => [1,5,3,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,5,3,4,2] => [1,5,2,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,5,4,2,3] => [1,5,4,2,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[1,5,4,3,2] => [1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[2,1,3,4,5] => [2,1,3,4,5] => [2,1,5,4,3] => [1,5,4,3,2] => 0

[2,1,3,5,4] => [2,1,3,5,4] => [2,1,5,4,3] => [1,5,4,3,2] => 0

[2,1,4,3,5] => [2,1,4,3,5] => [2,1,5,4,3] => [1,5,4,3,2] => 0

[2,1,4,5,3] => [2,1,3,5,4] => [2,1,5,4,3] => [1,5,4,3,2] => 0

[2,1,5,3,4] => [2,1,5,3,4] => [2,1,5,4,3] => [1,5,4,3,2] => 0

[2,1,5,4,3] => [2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,2] => 0

[2,3,1,4,5] => [1,3,2,4,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[2,3,1,5,4] => [1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[2,3,4,1,5] => [1,2,4,3,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[2,3,4,5,1] => [1,2,3,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[2,3,5,1,4] => [1,3,5,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[2,3,5,4,1] => [1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[2,4,1,3,5] => [2,4,1,3,5] => [2,5,1,4,3] => [1,5,4,2,3] => 1

[2,4,1,5,3] => [1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[2,4,3,1,5] => [1,4,3,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[2,4,3,5,1] => [1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[2,4,5,1,3] => [1,3,5,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[2,4,5,3,1] => [1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[2,5,1,3,4] => [2,5,1,3,4] => [2,5,1,4,3] => [1,5,4,2,3] => 1

[2,5,1,4,3] => [2,5,1,4,3] => [2,5,1,4,3] => [1,5,4,2,3] => 1

[2,5,3,1,4] => [1,5,3,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[2,5,3,4,1] => [1,5,2,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[2,5,4,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => [1,5,3,2,4] => 1

[2,5,4,3,1] => [1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[3,1,2,4,5] => [3,1,2,4,5] => [3,1,5,4,2] => [5,1,4,3,2] => 0

[3,1,2,5,4] => [3,1,2,5,4] => [3,1,5,4,2] => [5,1,4,3,2] => 0

[3,1,4,2,5] => [2,1,4,3,5] => [2,1,5,4,3] => [1,5,4,3,2] => 0

[3,1,4,5,2] => [2,1,3,5,4] => [2,1,5,4,3] => [1,5,4,3,2] => 0

[3,1,5,2,4] => [3,1,5,2,4] => [3,1,5,4,2] => [5,1,4,3,2] => 0

[3,1,5,4,2] => [2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,2] => 0

[3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,5,4] => [2,1,5,4,3] => 0

[3,2,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => [2,1,5,4,3] => 0

[3,2,4,1,5] => [2,1,4,3,5] => [2,1,5,4,3] => [1,5,4,3,2] => 0

[3,2,4,5,1] => [2,1,3,5,4] => [2,1,5,4,3] => [1,5,4,3,2] => 0

[3,2,5,1,4] => [3,2,5,1,4] => [3,2,5,1,4] => [2,1,5,3,4] => 1

[3,2,5,4,1] => [2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,2] => 0

[3,4,1,2,5] => [2,4,1,3,5] => [2,5,1,4,3] => [1,5,4,2,3] => 1

[3,4,1,5,2] => [1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[3,4,2,1,5] => [1,4,3,2,5] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[3,4,2,5,1] => [1,3,2,5,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[3,4,5,1,2] => [1,3,5,2,4] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[3,4,5,2,1] => [1,2,5,4,3] => [1,5,4,3,2] => [5,4,3,2,1] => 0

[3,5,1,2,4] => [3,5,1,2,4] => [3,5,1,4,2] => [5,1,4,2,3] => 0

[3,5,1,4,2] => [2,5,1,4,3] => [2,5,1,4,3] => [1,5,4,2,3] => 1

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Description

The number of occurrences of the vincular pattern |12-3 in a permutation.
This is the number of occurrences of the pattern $123$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly larger than the first entry of the permutation.

Map

Inverse fireworks map

Description

Sends a permutation to an inverse fireworks permutation.
A permutation $\sigma$ is inverse fireworks if its inverse avoids the vincular pattern $3-12$. The inverse fireworks map sends any permutation $\sigma$ to an inverse fireworks permutation that is below $\sigma$ in left weak order and has the same Rajchgot index St001759The Rajchgot index of a permutation..

Map

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].

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.