- FindStat

Identifier

Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000799: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 2$

$F_{3} = 2 + 4\ q$

$F_{4} = 4 + 8\ q + 12\ q^{2}$

$F_{5} = 12 + 24\ q + 36\ q^{2} + 48\ q^{3}$

$F_{6} = 48 + 96\ q + 144\ q^{2} + 192\ q^{3} + 240\ q^{4}$

Description

The number of occurrences of the vincular pattern |213 in a permutation.
This is the number of occurrences of the pattern

$(2,1,3)$ , such that the letter matched by

$2$ is the first entry of the permutation.

Map

cycle-as-one-line notation

Description

Return the permutation obtained by concatenating the cycles of a permutation, each written with minimal element first, sorted by minimal element.

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation

$\sigma$ of length

$n$ is the permutation

$\tau$ with

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

Map

Kreweras complement

Description

Sends the permutation

$\pi \in \mathfrak{S}_n$ to the permutation

$\pi^{-1}c$ where

$c = (1,\ldots,n)$ is the long cycle.