- FindStat

Identifier

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤ (values match St000318The number of addable cells of the Ferrers diagram of an integer partition., St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,0,1,0,1,0,1,0,0] => [5] => [[5],[]] => [] => 0

[1,1,0,1,0,1,1,0,0,0] => [5] => [[5],[]] => [] => 0

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

[1,1,0,1,1,0,0,1,0,0] => [5] => [[5],[]] => [] => 0

[1,1,0,1,1,0,1,0,0,0] => [5] => [[5],[]] => [] => 0

[1,1,0,1,1,1,0,0,0,0] => [5] => [[5],[]] => [] => 0

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

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

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

[1,1,1,0,0,1,0,1,0,0] => [5] => [[5],[]] => [] => 0

[1,1,1,0,0,1,1,0,0,0] => [5] => [[5],[]] => [] => 0

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

[1,1,1,0,1,0,0,1,0,0] => [5] => [[5],[]] => [] => 0

[1,1,1,0,1,0,1,0,0,0] => [5] => [[5],[]] => [] => 0

[1,1,1,0,1,1,0,0,0,0] => [5] => [[5],[]] => [] => 0

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

[1,1,1,1,0,0,0,1,0,0] => [5] => [[5],[]] => [] => 0

[1,1,1,1,0,0,1,0,0,0] => [5] => [[5],[]] => [] => 0

[1,1,1,1,0,1,0,0,0,0] => [5] => [[5],[]] => [] => 0

[1,1,1,1,1,0,0,0,0,0] => [5] => [[5],[]] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,1,0,1,0,1,0,1,0,0] => [1,5] => [[5,1],[]] => [] => 0

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,5] => [[5,1],[]] => [] => 0

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

[1,0,1,1,0,1,1,0,0,1,0,0] => [1,5] => [[5,1],[]] => [] => 0

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,5] => [[5,1],[]] => [] => 0

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,5] => [[5,1],[]] => [] => 0

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

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

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

[1,0,1,1,1,0,0,1,0,1,0,0] => [1,5] => [[5,1],[]] => [] => 0

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,5] => [[5,1],[]] => [] => 0

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

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,5] => [[5,1],[]] => [] => 0

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,5] => [[5,1],[]] => [] => 0

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,5] => [[5,1],[]] => [] => 0

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Description

The number of distinct parts of the integer partition.
This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.

Map

to ribbon

Description

The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.

Map

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.

Map

inner shape

Description

The inner shape of a skew partition.