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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Description
The number of missing boxes of a skew partition.
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.
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.
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.
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
rotate back to front
Description
The back to front rotation of an integer composition.
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