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Identifier

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000635: Posets ⟶ ℤ

Values

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

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$F_{2} = 2\ q$

$F_{3} = 3\ q + 3\ q^{4}$

Description

The number of strictly order preserving maps of a poset into itself.
A map

$f$ is strictly order preserving if

$a < b$ implies

$f(a) < f(b)$ .

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

pattern poset

Description

The pattern poset of a permutation.
This is the poset of all non-empty permutations that occur in the given permutation as a pattern, ordered by pattern containment.

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].