Identifier
-
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000579: Set partitions ⟶ ℤ
Values
[1,1] => [1,0,1,0] => {{1},{2}} => 1
[2] => [1,1,0,0] => {{1,2}} => 0
[1,1,1] => [1,0,1,0,1,0] => {{1},{2},{3}} => 3
[1,2] => [1,0,1,1,0,0] => {{1},{2,3}} => 1
[2,1] => [1,1,0,0,1,0] => {{1,2},{3}} => 2
[3] => [1,1,1,0,0,0] => {{1,2,3}} => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0] => {{1},{2},{3},{4}} => 6
[1,1,2] => [1,0,1,0,1,1,0,0] => {{1},{2},{3,4}} => 3
[1,2,1] => [1,0,1,1,0,0,1,0] => {{1},{2,3},{4}} => 4
[1,3] => [1,0,1,1,1,0,0,0] => {{1},{2,3,4}} => 1
[2,1,1] => [1,1,0,0,1,0,1,0] => {{1,2},{3},{4}} => 5
[2,2] => [1,1,0,0,1,1,0,0] => {{1,2},{3,4}} => 2
[3,1] => [1,1,1,0,0,0,1,0] => {{1,2,3},{4}} => 3
[4] => [1,1,1,1,0,0,0,0] => {{1,2,3,4}} => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => {{1},{2},{3},{4},{5}} => 10
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => {{1},{2},{3},{4,5}} => 6
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => {{1},{2},{3,4},{5}} => 7
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => {{1},{2},{3,4,5}} => 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => {{1},{2,3},{4},{5}} => 8
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => {{1},{2,3},{4,5}} => 4
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => {{1},{2,3,4},{5}} => 5
[1,4] => [1,0,1,1,1,1,0,0,0,0] => {{1},{2,3,4,5}} => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => {{1,2},{3},{4},{5}} => 9
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => {{1,2},{3},{4,5}} => 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => {{1,2},{3,4},{5}} => 6
[2,3] => [1,1,0,0,1,1,1,0,0,0] => {{1,2},{3,4,5}} => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => {{1,2,3},{4},{5}} => 7
[3,2] => [1,1,1,0,0,0,1,1,0,0] => {{1,2,3},{4,5}} => 3
[4,1] => [1,1,1,1,0,0,0,0,1,0] => {{1,2,3,4},{5}} => 4
[5] => [1,1,1,1,1,0,0,0,0,0] => {{1,2,3,4,5}} => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => {{1},{2},{3},{4},{5},{6}} => 15
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => {{1},{2},{3},{4},{5,6}} => 10
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => {{1},{2},{3},{4,5},{6}} => 11
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => {{1},{2},{3},{4,5,6}} => 6
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => {{1},{2},{3,4},{5},{6}} => 12
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => {{1},{2},{3,4},{5,6}} => 7
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => {{1},{2},{3,4,5},{6}} => 8
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => {{1},{2},{3,4,5,6}} => 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => {{1},{2,3},{4},{5},{6}} => 13
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => {{1},{2,3},{4},{5,6}} => 8
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => {{1},{2,3},{4,5},{6}} => 9
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => {{1},{2,3},{4,5,6}} => 4
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => {{1},{2,3,4},{5},{6}} => 10
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => {{1},{2,3,4},{5,6}} => 5
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => {{1},{2,3,4,5},{6}} => 6
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => {{1},{2,3,4,5,6}} => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => {{1,2},{3},{4},{5},{6}} => 14
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => {{1,2},{3},{4},{5,6}} => 9
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => {{1,2},{3},{4,5},{6}} => 10
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => {{1,2},{3},{4,5,6}} => 5
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => {{1,2},{3,4},{5},{6}} => 11
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => {{1,2},{3,4},{5,6}} => 6
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => {{1,2},{3,4,5},{6}} => 7
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => {{1,2},{3,4,5,6}} => 2
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => {{1,2,3},{4},{5},{6}} => 12
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => {{1,2,3},{4},{5,6}} => 7
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => {{1,2,3},{4,5},{6}} => 8
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => {{1,2,3},{4,5,6}} => 3
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => {{1,2,3,4},{5},{6}} => 9
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => {{1,2,3,4},{5,6}} => 4
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => {{1,2,3,4,5},{6}} => 5
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => {{1,2,3,4,5,6}} => 0
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => {{1},{2},{3},{4},{5},{6},{7}} => 21
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => {{1},{2},{3},{4},{5},{6,7}} => 15
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => {{1},{2},{3},{4},{5,6},{7}} => 16
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => {{1},{2},{3},{4},{5,6,7}} => 10
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => {{1},{2},{3},{4,5},{6},{7}} => 17
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => {{1},{2},{3},{4,5},{6,7}} => 11
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => {{1},{2},{3},{4,5,6},{7}} => 12
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => {{1},{2},{3},{4,5,6,7}} => 6
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => {{1},{2},{3,4},{5},{6},{7}} => 18
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => {{1},{2},{3,4},{5},{6,7}} => 12
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => {{1},{2},{3,4},{5,6},{7}} => 13
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => {{1},{2},{3,4},{5,6,7}} => 7
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => {{1},{2},{3,4,5},{6},{7}} => 14
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => {{1},{2},{3,4,5},{6,7}} => 8
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => {{1},{2},{3,4,5,6},{7}} => 9
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => {{1},{2},{3,4,5,6,7}} => 3
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => {{1},{2,3},{4},{5},{6},{7}} => 19
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => {{1},{2,3},{4},{5},{6,7}} => 13
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => {{1},{2,3},{4},{5,6},{7}} => 14
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => {{1},{2,3},{4},{5,6,7}} => 8
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => {{1},{2,3},{4,5},{6},{7}} => 15
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => {{1},{2,3},{4,5},{6,7}} => 9
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => {{1},{2,3},{4,5,6},{7}} => 10
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => {{1},{2,3},{4,5,6,7}} => 4
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => {{1},{2,3,4},{5},{6},{7}} => 16
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => {{1},{2,3,4},{5},{6,7}} => 10
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => {{1},{2,3,4},{5,6},{7}} => 11
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => {{1},{2,3,4},{5,6,7}} => 5
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => {{1},{2,3,4,5},{6},{7}} => 12
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => {{1},{2,3,4,5},{6,7}} => 6
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => {{1},{2,3,4,5,6},{7}} => 7
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => {{1},{2,3,4,5,6,7}} => 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => {{1,2},{3},{4},{5},{6},{7}} => 20
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => {{1,2},{3},{4},{5},{6,7}} => 14
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => {{1,2},{3},{4},{5,6},{7}} => 15
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => {{1,2},{3},{4},{5,6,7}} => 9
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => {{1,2},{3},{4,5},{6},{7}} => 16
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => {{1,2},{3},{4,5},{6,7}} => 10
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => {{1,2},{3},{4,5,6},{7}} => 11
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Description
The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element.
This is the number of pairs $i\lt j$ in different blocks such that $j$ is the maximal element of a block.
This is the number of pairs $i\lt j$ in different blocks such that $j$ is the maximal element of a block.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to noncrossing partition
Description
Biane's map to noncrossing set partitions.
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