Identifier
-
Mp00128:
Set partitions
—to composition⟶
Integer compositions
Mp00038: Integer compositions —reverse⟶ 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] => [1,1,2] => [[2,1,1],[]] => 0
{{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] => [1,1,2] => [[2,1,1],[]] => 0
{{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] => [1,2,1] => [[2,2,1],[1]] => 1
{{1,4},{2},{3}} => [2,1,1] => [1,1,2] => [[2,1,1],[]] => 0
{{1},{2,4},{3}} => [1,2,1] => [1,2,1] => [[2,2,1],[1]] => 1
{{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] => [1,1,3] => [[3,1,1],[]] => 0
{{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] => [1,1,3] => [[3,1,1],[]] => 0
{{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] => [1,1,3] => [[3,1,1],[]] => 0
{{1,2},{3,5},{4}} => [2,2,1] => [1,2,2] => [[3,2,1],[1]] => 1
{{1,2},{3},{4,5}} => [2,1,2] => [2,1,2] => [[3,2,2],[1,1]] => 2
{{1,2},{3},{4},{5}} => [2,1,1,1] => [1,1,1,2] => [[2,1,1,1],[]] => 0
{{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] => [1,1,3] => [[3,1,1],[]] => 0
{{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] => [1,1,3] => [[3,1,1],[]] => 0
{{1,3},{2,5},{4}} => [2,2,1] => [1,2,2] => [[3,2,1],[1]] => 1
{{1,3},{2},{4,5}} => [2,1,2] => [2,1,2] => [[3,2,2],[1,1]] => 2
{{1,3},{2},{4},{5}} => [2,1,1,1] => [1,1,1,2] => [[2,1,1,1],[]] => 0
{{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] => [1,3,1] => [[3,3,1],[2]] => 2
{{1,5},{2,3},{4}} => [2,2,1] => [1,2,2] => [[3,2,1],[1]] => 1
{{1},{2,3,5},{4}} => [1,3,1] => [1,3,1] => [[3,3,1],[2]] => 2
{{1},{2,3},{4,5}} => [1,2,2] => [2,2,1] => [[3,3,2],[2,1]] => 3
{{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] => [1,1,3] => [[3,1,1],[]] => 0
{{1,4},{2,5},{3}} => [2,2,1] => [1,2,2] => [[3,2,1],[1]] => 1
{{1,4},{2},{3,5}} => [2,1,2] => [2,1,2] => [[3,2,2],[1,1]] => 2
{{1,4},{2},{3},{5}} => [2,1,1,1] => [1,1,1,2] => [[2,1,1,1],[]] => 0
{{1,5},{2,4},{3}} => [2,2,1] => [1,2,2] => [[3,2,1],[1]] => 1
{{1},{2,4,5},{3}} => [1,3,1] => [1,3,1] => [[3,3,1],[2]] => 2
{{1},{2,4},{3,5}} => [1,2,2] => [2,2,1] => [[3,3,2],[2,1]] => 3
{{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] => [2,1,2] => [[3,2,2],[1,1]] => 2
{{1},{2,5},{3,4}} => [1,2,2] => [2,2,1] => [[3,3,2],[2,1]] => 3
{{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] => [1,2,1,1] => [[2,2,2,1],[1,1]] => 2
{{1,5},{2},{3},{4}} => [2,1,1,1] => [1,1,1,2] => [[2,1,1,1],[]] => 0
{{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] => [1,2,1,1] => [[2,2,2,1],[1,1]] => 2
{{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 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
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
reverse
Description
Return the reversal of a composition.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.
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