- FindStat

Identifier

Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000799: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2 + q$

$F_{3} = 9 + 6\ q + q^{2}$

$F_{4} = 64 + 48\ q + 12\ q^{2} + q^{3}$

$F_{5} = 565 + 500\ q + 150\ q^{2} + 20\ q^{3} + 61\ q^{4}$

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

to Dyck path

Description

Sends a parking function to its support Dyck word.

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

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.