- FindStat

Identifier

Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000971: Set partitions ⟶ ℤ

Values

[1] => [1,0] => {{1}} => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The smallest closer of a set partition.
A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers.
In other words, this is the smallest among the maximal elements of the blocks.

Map

to Dyck path

Description

Sends a parking function to its support Dyck word.

Map

to noncrossing partition

Description

Biane's map to noncrossing set partitions.