- FindStat

Identifier

Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000057: Standard tableaux ⟶ ℤ

Values

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

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

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

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

[1,1,1] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => 3

[1,1,2] => [1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 2

[1,2,1] => [1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 2

[2,1,1] => [1,1,0,1,0,0] => [[1,2,4],[3,5,6]] => 2

[1,1,3] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1

[1,3,1] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1

[3,1,1] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => 1

[1,2,2] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 1

[2,1,2] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 1

[2,2,1] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => 1

[1,2,3] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0

[1,3,2] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0

[2,1,3] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0

[2,3,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0

[3,1,2] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0

[3,2,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => 0

[1,1,1,1] => [1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => 6

[1,1,1,2] => [1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => 5

[1,1,2,1] => [1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => 5

[1,2,1,1] => [1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => 5

[2,1,1,1] => [1,1,1,0,1,0,0,0] => [[1,2,3,5],[4,6,7,8]] => 5

[1,1,1,3] => [1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => 4

[1,1,3,1] => [1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => 4

[1,3,1,1] => [1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => 4

[3,1,1,1] => [1,1,1,0,0,1,0,0] => [[1,2,3,6],[4,5,7,8]] => 4

[1,1,1,4] => [1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 3

[1,1,4,1] => [1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 3

[1,4,1,1] => [1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 3

[4,1,1,1] => [1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => 3

[1,1,2,2] => [1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => 4

[1,2,1,2] => [1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => 4

[1,2,2,1] => [1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => 4

[2,1,1,2] => [1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => 4

[2,1,2,1] => [1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => 4

[2,2,1,1] => [1,1,0,1,1,0,0,0] => [[1,2,4,5],[3,6,7,8]] => 4

[1,1,2,3] => [1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 3

[1,1,3,2] => [1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 3

[1,2,1,3] => [1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 3

[1,2,3,1] => [1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 3

[1,3,1,2] => [1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 3

[1,3,2,1] => [1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 3

[2,1,1,3] => [1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 3

[2,1,3,1] => [1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 3

[2,3,1,1] => [1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 3

[3,1,1,2] => [1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 3

[3,1,2,1] => [1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 3

[3,2,1,1] => [1,1,0,1,0,1,0,0] => [[1,2,4,6],[3,5,7,8]] => 3

[1,1,2,4] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2

[1,1,4,2] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2

[1,2,1,4] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2

[1,2,4,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2

[1,4,1,2] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2

[1,4,2,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2

[2,1,1,4] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2

[2,1,4,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2

[2,4,1,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2

[4,1,1,2] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2

[4,1,2,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2

[4,2,1,1] => [1,1,0,1,0,0,1,0] => [[1,2,4,7],[3,5,6,8]] => 2

[1,1,3,3] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 2

[1,3,1,3] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 2

[1,3,3,1] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 2

[3,1,1,3] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 2

[3,1,3,1] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 2

[3,3,1,1] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => 2

[1,1,3,4] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 1

[1,1,4,3] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 1

[1,3,1,4] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 1

[1,3,4,1] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 1

[1,4,1,3] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 1

[1,4,3,1] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 1

[3,1,1,4] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 1

[3,1,4,1] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 1

[3,4,1,1] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 1

[4,1,1,3] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 1

[4,1,3,1] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 1

[4,3,1,1] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => 1

[1,2,2,2] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => 3

[2,1,2,2] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => 3

[2,2,1,2] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => 3

[2,2,2,1] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => 3

[1,2,2,3] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2

[1,2,3,2] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2

[1,3,2,2] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2

[2,1,2,3] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2

[2,1,3,2] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2

[2,2,1,3] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2

[2,2,3,1] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2

[2,3,1,2] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2

[2,3,2,1] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2

[3,1,2,2] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2

[3,2,1,2] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2

[3,2,2,1] => [1,0,1,1,0,1,0,0] => [[1,3,4,6],[2,5,7,8]] => 2

[1,2,2,4] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 1

[1,2,4,2] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 1

[1,4,2,2] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 1

[2,1,2,4] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => 1

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Description

The Shynar inversion number of a standard tableau.
Shynar's inversion number is the number of inversion pairs in a standard Young tableau, where an inversion pair is defined as a pair of integers (x,y) such that y > x and y appears strictly southwest of x in the tableau.

Map

to two-row standard tableau

Description

Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.

Map

to Dyck path

Description

Sends a parking function to its support Dyck word.