- FindStat

Identifier

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001438: Skew partitions ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 7 + 6\ q + 2\ q^{2}$

$F_{5} = 11 + 17\ q + 17\ q^{2} + 6\ q^{3} + q^{4}$

Description

The number of missing boxes of a skew partition.

Map

rotate back to front

Description

The back to front rotation of an integer composition.

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

The integer composition of block sizes of a set partition.
For a set partition of

$\{1,2,\dots,n\}$ , this is the integer composition of

$n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.