Identifier
-
Mp00028:
Dyck paths
—reverse⟶
Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001062: Set partitions ⟶ ℤ
Values
[1,0] => [1,0] => [1,1,0,0] => {{1,2}} => 2
[1,0,1,0] => [1,0,1,0] => [1,1,0,1,0,0] => {{1,3},{2}} => 2
[1,1,0,0] => [1,1,0,0] => [1,1,1,0,0,0] => {{1,2,3}} => 3
[1,0,1,0,1,0] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => {{1,4},{2},{3}} => 2
[1,0,1,1,0,0] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => {{1,4},{2,3}} => 2
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => {{1,3,4},{2}} => 3
[1,1,0,1,0,0] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => {{1,2,4},{3}} => 3
[1,1,1,0,0,0] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => {{1,2,3,4}} => 4
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => {{1,5},{2},{3},{4}} => 2
[1,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => {{1,5},{2,3},{4}} => 2
[1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => {{1,5},{2},{3,4}} => 2
[1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => {{1,5},{2,4},{3}} => 2
[1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => {{1,5},{2,3,4}} => 3
[1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => {{1,4,5},{2},{3}} => 3
[1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => {{1,4,5},{2,3}} => 3
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => {{1,3,5},{2},{4}} => 3
[1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => {{1,2,5},{3},{4}} => 3
[1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => {{1,2,5},{3,4}} => 3
[1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => {{1,3,4,5},{2}} => 4
[1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => {{1,2,4,5},{3}} => 4
[1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => {{1,2,3,5},{4}} => 4
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => {{1,2,3,4,5}} => 5
[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => {{1,6},{2},{3},{4},{5}} => 2
[1,0,1,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => {{1,6},{2,3},{4},{5}} => 2
[1,0,1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => {{1,6},{2},{3,4},{5}} => 2
[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => {{1,6},{2,4},{3},{5}} => 2
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => {{1,6},{2,3,4},{5}} => 3
[1,0,1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => {{1,6},{2},{3},{4,5}} => 2
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => {{1,6},{2,3},{4,5}} => 2
[1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => {{1,6},{2},{3,5},{4}} => 2
[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => {{1,6},{2,5},{3},{4}} => 2
[1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => {{1,6},{2,5},{3,4}} => 2
[1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => {{1,6},{2},{3,4,5}} => 3
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => {{1,6},{2,4,5},{3}} => 3
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => {{1,6},{2,3,5},{4}} => 3
[1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => {{1,6},{2,3,4,5}} => 4
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => {{1,5,6},{2},{3},{4}} => 3
[1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => {{1,5,6},{2,3},{4}} => 3
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => {{1,5,6},{2},{3,4}} => 3
[1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => {{1,5,6},{2,4},{3}} => 3
[1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => {{1,5,6},{2,3,4}} => 3
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => {{1,4,6},{2},{3},{5}} => 3
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => {{1,4,6},{2,3},{5}} => 3
[1,1,0,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => {{1,3,6},{2},{4},{5}} => 3
[1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => {{1,2,6},{3},{4},{5}} => 3
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => {{1,2,6},{3,4},{5}} => 3
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => {{1,3,6},{2},{4,5}} => 3
[1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => {{1,2,6},{3},{4,5}} => 3
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => {{1,2,6},{3,5},{4}} => 3
[1,1,0,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => {{1,2,6},{3,4,5}} => 3
[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => {{1,4,5,6},{2},{3}} => 4
[1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => {{1,4,5,6},{2,3}} => 4
[1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,1,1,0,0,0,0] => {{1,3,5,6},{2},{4}} => 4
[1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,1,1,0,0,0,0] => {{1,2,5,6},{3},{4}} => 4
[1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => {{1,2,5,6},{3,4}} => 4
[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,1,0,1,0,0,0,0] => {{1,3,4,6},{2},{5}} => 4
[1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,1,0,1,0,0,0,0] => {{1,2,4,6},{3},{5}} => 4
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,1,0,1,0,0,0,0] => {{1,2,3,6},{4},{5}} => 4
[1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => {{1,2,3,6},{4,5}} => 4
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => {{1,3,4,5,6},{2}} => 5
[1,1,1,1,0,0,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => {{1,2,4,5,6},{3}} => 5
[1,1,1,1,0,0,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => {{1,2,3,5,6},{4}} => 5
[1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => {{1,2,3,4,6},{5}} => 5
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => {{1,2,3,4,5,6}} => 6
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => {{1,7},{2},{3},{4},{5},{6}} => 2
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => {{1,7},{2,3},{4},{5},{6}} => 2
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,1,0,0] => {{1,7},{2},{3,4},{5},{6}} => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => {{1,7},{2,4},{3},{5},{6}} => 2
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => {{1,7},{2,3,4},{5},{6}} => 3
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,1,0,0] => {{1,7},{2},{3},{4,5},{6}} => 2
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,1,0,0] => {{1,7},{2,3},{4,5},{6}} => 2
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,1,0,0] => {{1,7},{2},{3,5},{4},{6}} => 2
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => {{1,7},{2,5},{3},{4},{6}} => 2
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,1,0,0] => {{1,7},{2,5},{3,4},{6}} => 2
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,1,0,0] => {{1,7},{2},{3,4,5},{6}} => 3
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => {{1,7},{2,4,5},{3},{6}} => 3
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => {{1,7},{2,3,5},{4},{6}} => 3
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => {{1,7},{2,3,4,5},{6}} => 4
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,1,0,0,1,0,0] => {{1,7},{2},{3},{4},{5,6}} => 2
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,0,1,0,0] => {{1,7},{2,3},{4},{5,6}} => 2
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,1,0,0,1,0,0] => {{1,7},{2},{3,4},{5,6}} => 2
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => [1,1,1,0,1,0,0,1,1,0,0,1,0,0] => {{1,7},{2,4},{3},{5,6}} => 2
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,1,0,0] => {{1,7},{2,3,4},{5,6}} => 3
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,0,1,1,0,1,0,0,1,0,0] => {{1,7},{2},{3},{4,6},{5}} => 2
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,1,0,0,1,0,0] => {{1,7},{2,3},{4,6},{5}} => 2
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,1,0,0,1,0,0] => {{1,7},{2},{3,6},{4},{5}} => 2
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0] => {{1,7},{2,6},{3},{4},{5}} => 2
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,1,0,0,1,0,0] => {{1,7},{2,6},{3,4},{5}} => 2
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,1,1,0,0,1,0,0,1,0,0] => {{1,7},{2},{3,6},{4,5}} => 2
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => [1,1,1,0,1,1,0,0,1,0,0,1,0,0] => {{1,7},{2,6},{3},{4,5}} => 2
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0,1,0] => [1,1,1,1,0,1,0,0,1,0,0,1,0,0] => {{1,7},{2,6},{3,5},{4}} => 2
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0] => {{1,7},{2,6},{3,4,5}} => 3
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,1,0,0] => {{1,7},{2},{3},{4,5,6}} => 3
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,1,0,0] => {{1,7},{2,3},{4,5,6}} => 3
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,1,1,0,0,0,1,0,0] => {{1,7},{2},{3,5,6},{4}} => 3
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => [1,1,1,0,1,0,1,1,0,0,0,1,0,0] => {{1,7},{2,5,6},{3},{4}} => 3
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,1,0,0] => {{1,7},{2,5,6},{3,4}} => 3
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,1,1,0,1,0,0,0,1,0,0] => {{1,7},{2},{3,4,6},{5}} => 3
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0] => {{1,7},{2,4,6},{3},{5}} => 3
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0,1,0] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0] => {{1,7},{2,3,6},{4},{5}} => 3
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0] => {{1,7},{2,3,6},{4,5}} => 3
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Description
The maximal size of a block of a set partition.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
to noncrossing partition
Description
Biane's map to noncrossing set partitions.
Map
reverse
Description
The reversal of a Dyck path.
This is the Dyck path obtained by reading the path backwards.
This is the Dyck path obtained by reading the path backwards.
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