- FindStat

Identifier

Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000577: Set partitions ⟶ ℤ

Values

[1,0,1,0] => {{1},{2}} => 1

[1,1,0,0] => {{1,2}} => 0

[1,0,1,0,1,0] => {{1},{2},{3}} => 3

[1,0,1,1,0,0] => {{1},{2,3}} => 2

[1,1,0,0,1,0] => {{1,2},{3}} => 1

[1,1,0,1,0,0] => {{1,3},{2}} => 1

[1,1,1,0,0,0] => {{1,2,3}} => 0

[1,0,1,0,1,0,1,0] => {{1},{2},{3},{4}} => 6

[1,0,1,0,1,1,0,0] => {{1},{2},{3,4}} => 5

[1,0,1,1,0,0,1,0] => {{1},{2,3},{4}} => 4

[1,0,1,1,0,1,0,0] => {{1},{2,4},{3}} => 4

[1,0,1,1,1,0,0,0] => {{1},{2,3,4}} => 3

[1,1,0,0,1,0,1,0] => {{1,2},{3},{4}} => 3

[1,1,0,0,1,1,0,0] => {{1,2},{3,4}} => 2

[1,1,0,1,0,0,1,0] => {{1,3},{2},{4}} => 3

[1,1,0,1,0,1,0,0] => {{1,4},{2},{3}} => 3

[1,1,0,1,1,0,0,0] => {{1,3,4},{2}} => 2

[1,1,1,0,0,0,1,0] => {{1,2,3},{4}} => 1

[1,1,1,0,0,1,0,0] => {{1,4},{2,3}} => 1

[1,1,1,0,1,0,0,0] => {{1,2,4},{3}} => 1

[1,1,1,1,0,0,0,0] => {{1,2,3,4}} => 0

[1,0,1,0,1,0,1,0,1,0] => {{1},{2},{3},{4},{5}} => 10

[1,0,1,0,1,0,1,1,0,0] => {{1},{2},{3},{4,5}} => 9

[1,0,1,0,1,1,0,0,1,0] => {{1},{2},{3,4},{5}} => 8

[1,0,1,0,1,1,0,1,0,0] => {{1},{2},{3,5},{4}} => 8

[1,0,1,0,1,1,1,0,0,0] => {{1},{2},{3,4,5}} => 7

[1,0,1,1,0,0,1,0,1,0] => {{1},{2,3},{4},{5}} => 7

[1,0,1,1,0,0,1,1,0,0] => {{1},{2,3},{4,5}} => 6

[1,0,1,1,0,1,0,0,1,0] => {{1},{2,4},{3},{5}} => 7

[1,0,1,1,0,1,0,1,0,0] => {{1},{2,5},{3},{4}} => 7

[1,0,1,1,0,1,1,0,0,0] => {{1},{2,4,5},{3}} => 6

[1,0,1,1,1,0,0,0,1,0] => {{1},{2,3,4},{5}} => 5

[1,0,1,1,1,0,0,1,0,0] => {{1},{2,5},{3,4}} => 5

[1,0,1,1,1,0,1,0,0,0] => {{1},{2,3,5},{4}} => 5

[1,0,1,1,1,1,0,0,0,0] => {{1},{2,3,4,5}} => 4

[1,1,0,0,1,0,1,0,1,0] => {{1,2},{3},{4},{5}} => 6

[1,1,0,0,1,0,1,1,0,0] => {{1,2},{3},{4,5}} => 5

[1,1,0,0,1,1,0,0,1,0] => {{1,2},{3,4},{5}} => 4

[1,1,0,0,1,1,0,1,0,0] => {{1,2},{3,5},{4}} => 4

[1,1,0,0,1,1,1,0,0,0] => {{1,2},{3,4,5}} => 3

[1,1,0,1,0,0,1,0,1,0] => {{1,3},{2},{4},{5}} => 6

[1,1,0,1,0,0,1,1,0,0] => {{1,3},{2},{4,5}} => 5

[1,1,0,1,0,1,0,0,1,0] => {{1,4},{2},{3},{5}} => 6

[1,1,0,1,0,1,0,1,0,0] => {{1,5},{2},{3},{4}} => 6

[1,1,0,1,0,1,1,0,0,0] => {{1,4,5},{2},{3}} => 5

[1,1,0,1,1,0,0,0,1,0] => {{1,3,4},{2},{5}} => 4

[1,1,0,1,1,0,0,1,0,0] => {{1,5},{2},{3,4}} => 4

[1,1,0,1,1,0,1,0,0,0] => {{1,3,5},{2},{4}} => 4

[1,1,0,1,1,1,0,0,0,0] => {{1,3,4,5},{2}} => 3

[1,1,1,0,0,0,1,0,1,0] => {{1,2,3},{4},{5}} => 3

[1,1,1,0,0,0,1,1,0,0] => {{1,2,3},{4,5}} => 2

[1,1,1,0,0,1,0,0,1,0] => {{1,4},{2,3},{5}} => 3

[1,1,1,0,0,1,0,1,0,0] => {{1,5},{2,3},{4}} => 3

[1,1,1,0,0,1,1,0,0,0] => {{1,4,5},{2,3}} => 2

[1,1,1,0,1,0,0,0,1,0] => {{1,2,4},{3},{5}} => 3

[1,1,1,0,1,0,0,1,0,0] => {{1,5},{2,4},{3}} => 3

[1,1,1,0,1,0,1,0,0,0] => {{1,2,5},{3},{4}} => 3

[1,1,1,0,1,1,0,0,0,0] => {{1,2,4,5},{3}} => 2

[1,1,1,1,0,0,0,0,1,0] => {{1,2,3,4},{5}} => 1

[1,1,1,1,0,0,0,1,0,0] => {{1,5},{2,3,4}} => 1

[1,1,1,1,0,0,1,0,0,0] => {{1,2,5},{3,4}} => 1

[1,1,1,1,0,1,0,0,0,0] => {{1,2,3,5},{4}} => 1

[1,1,1,1,1,0,0,0,0,0] => {{1,2,3,4,5}} => 0

[1,0,1,0,1,0,1,0,1,0,1,0] => {{1},{2},{3},{4},{5},{6}} => 15

[1,0,1,0,1,0,1,0,1,1,0,0] => {{1},{2},{3},{4},{5,6}} => 14

[1,0,1,0,1,0,1,1,0,0,1,0] => {{1},{2},{3},{4,5},{6}} => 13

[1,0,1,0,1,0,1,1,0,1,0,0] => {{1},{2},{3},{4,6},{5}} => 13

[1,0,1,0,1,0,1,1,1,0,0,0] => {{1},{2},{3},{4,5,6}} => 12

[1,0,1,0,1,1,0,0,1,0,1,0] => {{1},{2},{3,4},{5},{6}} => 12

[1,0,1,0,1,1,0,0,1,1,0,0] => {{1},{2},{3,4},{5,6}} => 11

[1,0,1,0,1,1,0,1,0,0,1,0] => {{1},{2},{3,5},{4},{6}} => 12

[1,0,1,0,1,1,0,1,0,1,0,0] => {{1},{2},{3,6},{4},{5}} => 12

[1,0,1,0,1,1,0,1,1,0,0,0] => {{1},{2},{3,5,6},{4}} => 11

[1,0,1,0,1,1,1,0,0,0,1,0] => {{1},{2},{3,4,5},{6}} => 10

[1,0,1,0,1,1,1,0,0,1,0,0] => {{1},{2},{3,6},{4,5}} => 10

[1,0,1,0,1,1,1,0,1,0,0,0] => {{1},{2},{3,4,6},{5}} => 10

[1,0,1,0,1,1,1,1,0,0,0,0] => {{1},{2},{3,4,5,6}} => 9

[1,0,1,1,0,0,1,0,1,0,1,0] => {{1},{2,3},{4},{5},{6}} => 11

[1,0,1,1,0,0,1,0,1,1,0,0] => {{1},{2,3},{4},{5,6}} => 10

[1,0,1,1,0,0,1,1,0,0,1,0] => {{1},{2,3},{4,5},{6}} => 9

[1,0,1,1,0,0,1,1,0,1,0,0] => {{1},{2,3},{4,6},{5}} => 9

[1,0,1,1,0,0,1,1,1,0,0,0] => {{1},{2,3},{4,5,6}} => 8

[1,0,1,1,0,1,0,0,1,0,1,0] => {{1},{2,4},{3},{5},{6}} => 11

[1,0,1,1,0,1,0,0,1,1,0,0] => {{1},{2,4},{3},{5,6}} => 10

[1,0,1,1,0,1,0,1,0,0,1,0] => {{1},{2,5},{3},{4},{6}} => 11

[1,0,1,1,0,1,0,1,0,1,0,0] => {{1},{2,6},{3},{4},{5}} => 11

[1,0,1,1,0,1,0,1,1,0,0,0] => {{1},{2,5,6},{3},{4}} => 10

[1,0,1,1,0,1,1,0,0,0,1,0] => {{1},{2,4,5},{3},{6}} => 9

[1,0,1,1,0,1,1,0,0,1,0,0] => {{1},{2,6},{3},{4,5}} => 9

[1,0,1,1,0,1,1,0,1,0,0,0] => {{1},{2,4,6},{3},{5}} => 9

[1,0,1,1,0,1,1,1,0,0,0,0] => {{1},{2,4,5,6},{3}} => 8

[1,0,1,1,1,0,0,0,1,0,1,0] => {{1},{2,3,4},{5},{6}} => 8

[1,0,1,1,1,0,0,0,1,1,0,0] => {{1},{2,3,4},{5,6}} => 7

[1,0,1,1,1,0,0,1,0,0,1,0] => {{1},{2,5},{3,4},{6}} => 8

[1,0,1,1,1,0,0,1,0,1,0,0] => {{1},{2,6},{3,4},{5}} => 8

[1,0,1,1,1,0,0,1,1,0,0,0] => {{1},{2,5,6},{3,4}} => 7

[1,0,1,1,1,0,1,0,0,0,1,0] => {{1},{2,3,5},{4},{6}} => 8

[1,0,1,1,1,0,1,0,0,1,0,0] => {{1},{2,6},{3,5},{4}} => 8

[1,0,1,1,1,0,1,0,1,0,0,0] => {{1},{2,3,6},{4},{5}} => 8

[1,0,1,1,1,0,1,1,0,0,0,0] => {{1},{2,3,5,6},{4}} => 7

[1,0,1,1,1,1,0,0,0,0,1,0] => {{1},{2,3,4,5},{6}} => 6

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Description

The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element.
This is the number of pairs $i\lt j$ in different blocks such that $i$ is the maximal element of a block.

Map

to noncrossing partition

Description

Biane's map to noncrossing set partitions.