- FindStat

Identifier

Mp00053: Parking functions —to car permutation⟶ Permutations
Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an ascent if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).

Map

to car permutation

Description

Maps a parking function to the permutation of cars that correspond to that parking function.
For example, the image $[2, 4, 5, 6, 3, 1, 7]$ means that car 2 takes spots 1, car 4 takes spot 2, and so on.

Map

Robinson-Schensted insertion tableau

Description

Sends a permutation to its Robinson-Schensted insertion tableau.
The Robinson-Schensted corrspondence is a bijection between permutations of length $n$ and pairs of standard Young tableaux of the same shape and of size $n$, see [1]. These two tableaux are the insertion tableau and the recording tableau.
This map sends a permutation to its corresponding insertion tableau.