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Identifier

Mp00164: Set partitions —Chen Deng Du Stanley Yan⟶ Set partitions
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00308: Integer partitions —Bulgarian solitaire⟶ Integer partitions
St000010: Integer partitions ⟶ ℤ

Values

{{1}} => {{1}} => [1] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,5,6}} => {{1,2,3,4,5,6}} => [6] => [5,1] => 2

{{1,2,3,4,5},{6}} => {{1,2,3,4,5},{6}} => [5,1] => [4,2] => 2

{{1,2,3,4,6},{5}} => {{1,2,3,4,6},{5}} => [5,1] => [4,2] => 2

{{1,2,3,4},{5,6}} => {{1,2,3,4},{5,6}} => [4,2] => [3,2,1] => 3

{{1,2,3,4},{5},{6}} => {{1,2,3,4},{5},{6}} => [4,1,1] => [3,3] => 2

{{1,2,3,5,6},{4}} => {{1,2,3,5,6},{4}} => [5,1] => [4,2] => 2

{{1,2,3,5},{4,6}} => {{1,2,3,6},{4,5}} => [4,2] => [3,2,1] => 3

{{1,2,3,5},{4},{6}} => {{1,2,3,5},{4},{6}} => [4,1,1] => [3,3] => 2

{{1,2,3,6},{4,5}} => {{1,2,3,5},{4,6}} => [4,2] => [3,2,1] => 3

{{1,2,3},{4,5,6}} => {{1,2,3},{4,5,6}} => [3,3] => [2,2,2] => 3

{{1,2,3},{4,5},{6}} => {{1,2,3},{4,5},{6}} => [3,2,1] => [3,2,1] => 3

{{1,2,3,6},{4},{5}} => {{1,2,3,6},{4},{5}} => [4,1,1] => [3,3] => 2

{{1,2,3},{4,6},{5}} => {{1,2,3},{4,6},{5}} => [3,2,1] => [3,2,1] => 3

{{1,2,3},{4},{5,6}} => {{1,2,3},{4},{5,6}} => [3,2,1] => [3,2,1] => 3

{{1,2,3},{4},{5},{6}} => {{1,2,3},{4},{5},{6}} => [3,1,1,1] => [4,2] => 2

{{1,2,4,5,6},{3}} => {{1,2,4,5,6},{3}} => [5,1] => [4,2] => 2

{{1,2,4,5},{3,6}} => {{1,2,5},{3,4,6}} => [3,3] => [2,2,2] => 3

{{1,2,4,5},{3},{6}} => {{1,2,4,5},{3},{6}} => [4,1,1] => [3,3] => 2

{{1,2,4,6},{3,5}} => {{1,2,6},{3,4,5}} => [3,3] => [2,2,2] => 3

{{1,2,4},{3,5,6}} => {{1,2,5,6},{3,4}} => [4,2] => [3,2,1] => 3

{{1,2,4},{3,5},{6}} => {{1,2,5},{3,4},{6}} => [3,2,1] => [3,2,1] => 3

{{1,2,4,6},{3},{5}} => {{1,2,4,6},{3},{5}} => [4,1,1] => [3,3] => 2

{{1,2,4},{3,6},{5}} => {{1,2,6},{3,4},{5}} => [3,2,1] => [3,2,1] => 3

{{1,2,4},{3},{5,6}} => {{1,2,4},{3},{5,6}} => [3,2,1] => [3,2,1] => 3

{{1,2,4},{3},{5},{6}} => {{1,2,4},{3},{5},{6}} => [3,1,1,1] => [4,2] => 2

{{1,2,5,6},{3,4}} => {{1,2,4},{3,5,6}} => [3,3] => [2,2,2] => 3

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Description

The length of the partition.

Map

Chen Deng Du Stanley Yan

Description

A map that swaps the crossing number and the nesting number of a set partition.

Map

Bulgarian solitaire

Description

A move in Bulgarian solitaire.
Remove the first column of the Ferrers diagram and insert it as a new row.
If the partition is empty, return the empty partition.

Map

shape

Description

Sends a set partition to the integer partition obtained by the sizes of the blocks.