- FindStat

Identifier

Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000800: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1441 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The number of occurrences of the vincular pattern |231 in a permutation.
This is the number of occurrences of the pattern $(2,3,1)$, 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

Ringel

Description

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