- FindStat

Identifier

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001729: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1525 entries. <<<

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 5\ q$

$F_{4} = 1 + 20\ q + 3\ q^{2}$

$F_{5} = 1 + 84\ q + 35\ q^{2}$

$F_{6} = 1 + 409\ q + 295\ q^{2} + 15\ q^{3}$

Description

The number of visible descents of a permutation.
A visible descent of a permutation

$\pi$ is a position

$i$ such that

$\pi(i+1) \leq \min(i, \pi(i))$ .

Map

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation

$\sigma$ of length

$n$ is given by

$\tau$ with

$\tau(i) = \sigma(n+1-i)$ .

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].