Identifier
Values
[1] => [1,0] => {{1}} => 0
[2] => [1,0,1,0] => {{1},{2}} => 0
[1,1] => [1,1,0,0] => {{1,2}} => 1
[3] => [1,0,1,0,1,0] => {{1},{2},{3}} => 0
[2,1] => [1,0,1,1,0,0] => {{1},{2,3}} => 1
[1,1,1] => [1,1,0,1,0,0] => {{1,3},{2}} => 1
[4] => [1,0,1,0,1,0,1,0] => {{1},{2},{3},{4}} => 0
[3,1] => [1,0,1,0,1,1,0,0] => {{1},{2},{3,4}} => 1
[2,2] => [1,1,1,0,0,0] => {{1,2,3}} => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => {{1},{2,4},{3}} => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => {{1,4},{2},{3}} => 1
[5] => [1,0,1,0,1,0,1,0,1,0] => {{1},{2},{3},{4},{5}} => 0
[4,1] => [1,0,1,0,1,0,1,1,0,0] => {{1},{2},{3},{4,5}} => 1
[3,2] => [1,0,1,1,1,0,0,0] => {{1},{2,3,4}} => 2
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => {{1},{2},{3,5},{4}} => 1
[2,2,1] => [1,1,1,0,0,1,0,0] => {{1,4},{2,3}} => 2
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => {{1},{2,5},{3},{4}} => 1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => {{1,5},{2},{3},{4}} => 1
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => {{1},{2},{3},{4},{5},{6}} => 0
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => {{1},{2},{3},{4},{5,6}} => 1
[4,2] => [1,0,1,0,1,1,1,0,0,0] => {{1},{2},{3,4,5}} => 2
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => {{1},{2},{3},{4,6},{5}} => 1
[3,3] => [1,1,1,0,1,0,0,0] => {{1,2,4},{3}} => 2
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => {{1},{2,5},{3,4}} => 2
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => {{1},{2},{3,6},{4},{5}} => 1
[2,2,2] => [1,1,1,1,0,0,0,0] => {{1,2,3,4}} => 3
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => {{1,5},{2,3},{4}} => 2
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => {{1},{2,6},{3},{4},{5}} => 1
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => {{1,6},{2},{3},{4},{5}} => 1
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => {{1},{2},{3},{4},{5},{6},{7}} => 0
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => {{1},{2},{3},{4},{5},{6,7}} => 1
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => {{1},{2},{3},{4,5,6}} => 2
[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => {{1},{2},{3},{4},{5,7},{6}} => 1
[4,3] => [1,0,1,1,1,0,1,0,0,0] => {{1},{2,3,5},{4}} => 2
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => {{1},{2},{3,6},{4,5}} => 2
[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => {{1},{2},{3},{4,7},{5},{6}} => 1
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => {{1,5},{2,4},{3}} => 2
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => {{1},{2,3,4,5}} => 3
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => {{1},{2,6},{3,4},{5}} => 2
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => {{1},{2},{3,7},{4},{5},{6}} => 1
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => {{1,5},{2,3,4}} => 3
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => {{1,6},{2,3},{4},{5}} => 2
[2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => {{1},{2,7},{3},{4},{5},{6}} => 1
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => {{1,7},{2},{3},{4},{5},{6}} => 1
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => {{1},{2},{3},{4},{5,6,7}} => 2
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => {{1},{2},{3,4,6},{5}} => 2
[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => {{1},{2},{3},{4,7},{5,6}} => 2
[4,4] => [1,1,1,0,1,0,1,0,0,0] => {{1,2,5},{3},{4}} => 2
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => {{1},{2,6},{3,5},{4}} => 2
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => {{1},{2},{3,4,5,6}} => 3
[4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => {{1},{2},{3,7},{4,5},{6}} => 2
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => {{1,2,4,5},{3}} => 3
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => {{1,6},{2,4},{3},{5}} => 2
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => {{1},{2,6},{3,4,5}} => 3
[3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => {{1},{2,7},{3,4},{5},{6}} => 2
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => {{1,2,3,5},{4}} => 3
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => {{1,6},{2,3,4},{5}} => 3
[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => {{1,7},{2,3},{4},{5},{6}} => 2
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => {{1},{2},{3},{4,5,7},{6}} => 2
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => {{1},{2,3,6},{4},{5}} => 2
[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => {{1},{2},{3,7},{4,6},{5}} => 2
[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => {{1},{2},{3},{4,5,6,7}} => 3
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => {{1,6},{2,5},{3},{4}} => 2
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => {{1},{2,3,5,6},{4}} => 3
[4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => {{1},{2,7},{3,5},{4},{6}} => 2
[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => {{1},{2},{3,7},{4,5,6}} => 3
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => {{1,2,3,4,5}} => 4
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => {{1,6},{2,4,5},{3}} => 3
[3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => {{1,7},{2,4},{3},{5},{6}} => 2
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => {{1},{2,3,4,6},{5}} => 3
[3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => {{1},{2,7},{3,4,5},{6}} => 3
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => {{1,6},{2,3,5},{4}} => 3
[2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => {{1,7},{2,3,4},{5},{6}} => 3
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => {{1},{2},{3,4,7},{5},{6}} => 2
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => {{1,2,6},{3},{4},{5}} => 2
[5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => {{1},{2,7},{3,6},{4},{5}} => 2
[5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => {{1},{2},{3,4,6,7},{5}} => 3
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => {{1,2,5,6},{3},{4}} => 3
[4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => {{1,7},{2,5},{3},{4},{6}} => 2
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => {{1},{2,3,4,5,6}} => 4
[4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => {{1},{2,7},{3,5,6},{4}} => 3
[4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => {{1},{2},{3,4,5,7},{6}} => 3
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => {{1,6},{2,3,4,5}} => 4
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => {{1,2,4,6},{3},{5}} => 3
[3,3,2,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => {{1,7},{2,4,5},{3},{6}} => 3
[3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => {{1},{2,7},{3,4,6},{5}} => 3
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => {{1,2,3,6},{4},{5}} => 3
[2,2,2,2,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => {{1,7},{2,3,5},{4},{6}} => 3
[6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => {{1},{2,3,7},{4},{5},{6}} => 2
[5,5,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0] => {{1,7},{2,6},{3},{4},{5}} => 2
[5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => {{1},{2,3,6,7},{4},{5}} => 3
[5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => {{1},{2},{3,4,5,6,7}} => 4
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => {{1,2,4,5,6},{3}} => 4
[4,4,2,1] => [1,1,1,0,1,0,1,1,0,0,0,1,0,0] => {{1,7},{2,5,6},{3},{4}} => 3
[4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => {{1},{2,7},{3,4,5,6}} => 4
[4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => {{1},{2,3,5,7},{4},{6}} => 3
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => {{1,2,3,6},{4,5}} => 4
[3,3,3,1,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => {{1,7},{2,3,4,5},{6}} => 4
[3,3,2,2,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0] => {{1,7},{2,4,6},{3},{5}} => 3
[3,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => {{1},{2,3,4,7},{5},{6}} => 3
[2,2,2,2,2,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0] => {{1,7},{2,3,6},{4},{5}} => 3
>>> Load all 127 entries. <<<
[6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => {{1,2,7},{3},{4},{5},{6}} => 2
[5,5,2] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => {{1,2,6,7},{3},{4},{5}} => 3
[5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => {{1},{2,3,5,6,7},{4}} => 4
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => {{1,2,3,4,6},{5}} => 4
[4,4,3,1] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0] => {{1,7},{2,4,5,6},{3}} => 4
[4,4,2,2] => [1,1,1,0,1,0,1,1,0,1,0,0,0,0] => {{1,2,5,7},{3},{4},{6}} => 3
[4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => {{1},{2,3,4,7},{5,6}} => 4
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => {{1,2,3,4,5,6}} => 5
[3,3,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0] => {{1,7},{2,3,6},{4,5}} => 4
[3,3,2,2,2] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0] => {{1,2,4,7},{3},{5},{6}} => 3
[2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0] => {{1,2,3,7},{4},{5},{6}} => 3
[5,5,3] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0] => {{1,2,5,6,7},{3},{4}} => 4
[5,4,4] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => {{1},{2,3,4,5,7},{6}} => 4
[4,4,4,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0] => {{1,7},{2,3,4,6},{5}} => 4
[4,4,3,2] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => {{1,2,4,7},{3},{5,6}} => 4
[4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => {{1},{2,3,4,5,6,7}} => 5
[3,3,3,3,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => {{1,7},{2,3,4,5,6}} => 5
[3,3,3,2,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0] => {{1,2,3,7},{4,5},{6}} => 4
[5,5,4] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0] => {{1,2,4,5,7},{3},{6}} => 4
[4,4,4,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0] => {{1,2,3,7},{4,6},{5}} => 4
[4,4,3,3] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => {{1,2,4,5,6,7},{3}} => 5
[3,3,3,3,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => {{1,2,3,7},{4,5,6}} => 5
[5,5,5] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => {{1,2,3,4,7},{5},{6}} => 4
[4,4,4,3] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => {{1,2,3,4,6,7},{5}} => 5
[3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => {{1,2,3,4,5,7},{6}} => 5
[4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => {{1,2,3,4,5,6,7}} => 6
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Description
The rank of the set partition.
This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to noncrossing partition
Description
Biane's map to noncrossing set partitions.