- FindStat

Identifier

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000124: Permutations ⟶ ℤ

Values

[1] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 5712 entries. <<<

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Description

The cardinality of the preimage of the Simion-Schmidt map.
The Simion-Schmidt bijection transforms a [3,1,2]-avoiding permutation into a [3,2,1]-avoiding permutation. More generally, it can be thought of as a map $S$ that turns any permutation into a [3,2,1]-avoiding permutation. This statistic is the size of $S^{-1}(\pi)$ for each permutation $\pi$.
The map $S$ can also be realized using the quotient of the $0$-Hecke Monoid of the symmetric group by the relation $\pi_i \pi_{i+1} \pi_i = \pi_{i+1} \pi_i$, sending each element of the fiber of the quotient to the unique [3,2,1]-avoiding element in that fiber.

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].