- FindStat

Identifier

Mp00287: Ordered set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001435: Skew partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of missing boxes in the first row.

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

to composition

Description

Return the sizes of the blocks as a composition.