- FindStat

Identifier

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00220: Set partitions —Yip⟶ Set partitions
St000492: Set partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => {{1,2,3,4,5},{6}} => {{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}} => {{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}} => {{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}} => {{1,7},{2},{3},{4},{5},{6}} => 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}} => {{1,6},{2},{3},{4},{5},{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}} => {{1,6,7},{2},{3},{4},{5}} => 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}} => {{1,5},{2},{3},{4},{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}} => {{1,5,7},{2},{3},{4},{6}} => 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}} => {{1,5,6},{2},{3},{4},{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}} => {{1,5,6,7},{2},{3},{4}} => 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}} => {{1,4},{2},{3},{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}} => {{1,4,7},{2},{3},{5},{6}} => 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}} => {{1,4,6},{2},{3},{5},{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}} => {{1,4,6,7},{2},{3},{5}} => 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}} => {{1,4,5},{2},{3},{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}} => {{1,4,5,7},{2},{3},{6}} => 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}} => {{1,4,5,6},{2},{3},{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}} => {{1,4,5,6,7},{2},{3}} => 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}} => {{1,3},{2},{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}} => {{1,3,7},{2},{4},{5},{6}} => 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}} => {{1,3,6},{2},{4},{5},{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}} => {{1,3,6,7},{2},{4},{5}} => 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}} => {{1,3,5},{2},{4},{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}} => {{1,3,5,7},{2},{4},{6}} => 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}} => {{1,3,5,6},{2},{4},{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}} => {{1,3,5,6,7},{2},{4}} => 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}} => {{1,3,4},{2},{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}} => {{1,3,4,7},{2},{5},{6}} => 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}} => {{1,3,4,6},{2},{5},{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}} => {{1,3,4,6,7},{2},{5}} => 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}} => {{1,3,4,5},{2},{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}} => {{1,3,4,5,7},{2},{6}} => 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}} => {{1,3,4,5,6},{2},{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}} => {{1,3,4,5,6,7},{2}} => 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}} => {{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}} => {{1,2,7},{3},{4},{5},{6}} => 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}} => {{1,2,6},{3},{4},{5},{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}} => {{1,2,6,7},{3},{4},{5}} => 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}} => {{1,2,5},{3},{4},{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}} => {{1,2,5,7},{3},{4},{6}} => 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}} => {{1,2,5,6},{3},{4},{7}} => 11

>>> Load all 136 entries. <<<

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

[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => {{1,2},{3,4},{5},{6},{7}} => {{1,2,4},{3},{5},{6},{7}} => 17

[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => {{1,2},{3,4},{5},{6,7}} => {{1,2,4,7},{3},{5},{6}} => 11

[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => {{1,2},{3,4},{5,6},{7}} => {{1,2,4,6},{3},{5},{7}} => 12

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

[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => {{1,2},{3,4,5},{6},{7}} => {{1,2,4,5},{3},{6},{7}} => 13

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

[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => {{1,2},{3,4,5,6},{7}} => {{1,2,4,5,6},{3},{7}} => 8

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

[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => {{1,2,3},{4},{5},{6},{7}} => {{1,2,3},{4},{5},{6},{7}} => 18

[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => {{1,2,3},{4},{5},{6,7}} => {{1,2,3,7},{4},{5},{6}} => 12

[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => {{1,2,3},{4},{5,6},{7}} => {{1,2,3,6},{4},{5},{7}} => 13

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

[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => {{1,2,3},{4,5},{6},{7}} => {{1,2,3,5},{4},{6},{7}} => 14

[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => {{1,2,3},{4,5},{6,7}} => {{1,2,3,5,7},{4},{6}} => 8

[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => {{1,2,3},{4,5,6},{7}} => {{1,2,3,5,6},{4},{7}} => 9

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

[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => {{1,2,3,4},{5},{6},{7}} => {{1,2,3,4},{5},{6},{7}} => 15

[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => {{1,2,3,4},{5},{6,7}} => {{1,2,3,4,7},{5},{6}} => 9

[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => {{1,2,3,4},{5,6},{7}} => {{1,2,3,4,6},{5},{7}} => 10

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

[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => {{1,2,3,4,5},{6},{7}} => {{1,2,3,4,5},{6},{7}} => 11

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

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

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

[1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => {{1},{2},{3,4,5,6,7,8}} => {{1,4,5,6,7,8},{2},{3}} => 3

[1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0] => {{1},{2,3},{4,5,6,7,8}} => {{1,3,5,6,7,8},{2},{4}} => 4

[1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => {{1},{2,3,4,5,6,7,8}} => {{1,3,4,5,6,7,8},{2}} => 1

[2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => {{1,2},{3,4,5,6,7,8}} => {{1,2,4,5,6,7,8},{3}} => 2

[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0] => {{1,2,3},{4,5,6,7,8}} => {{1,2,3,5,6,7,8},{4}} => 3

[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => {{1,2,3,4},{5,6,7,8}} => {{1,2,3,4,6,7,8},{5}} => 4

[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0] => {{1,2,3,4,5},{6,7,8}} => {{1,2,3,4,5,7,8},{6}} => 5

[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => {{1,2,3,4,5,6},{7,8}} => {{1,2,3,4,5,6,8},{7}} => 6

[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => {{1,2,3,4,5,6,7},{8}} => {{1,2,3,4,5,6,7},{8}} => 7

[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => {{1,2,3,4,5,6,7,8}} => {{1,2,3,4,5,6,7,8}} => 0

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$F_{2} = 1 + q$

$F_{3} = 1 + q + q^{2} + q^{3}$

$F_{4} = 1 + q + q^{2} + 2\ q^{3} + q^{4} + q^{5} + q^{6}$

$F_{5} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + 2\ q^{5} + 2\ q^{6} + 2\ q^{7} + q^{8} + q^{9} + q^{10}$

$F_{6} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 3\ q^{6} + 3\ q^{7} + 3\ q^{8} + 3\ q^{9} + 3\ q^{10} + 2\ q^{11} + 2\ q^{12} + q^{13} + q^{14} + q^{15}$

$F_{7} = 1 + q + q^{2} + 2\ q^{3} + 2\ q^{4} + 3\ q^{5} + 4\ q^{6} + 4\ q^{7} + 4\ q^{8} + 5\ q^{9} + 5\ q^{10} + 5\ q^{11} + 5\ q^{12} + 4\ q^{13} + 4\ q^{14} + 4\ q^{15} + 3\ q^{16} + 2\ q^{17} + 2\ q^{18} + q^{19} + q^{20} + q^{21}$

Description

The rob statistic of a set partition.
Let

$S = B_1,\ldots,B_k$ be a set partition with ordered blocks

$B_i$ and with

$\operatorname{min} B_a < \operatorname{min} B_b$ for

$a < b$ .
According to [1, Definition 3], a rob (right-opener-bigger) of

$S$ is given by a pair

$i < j$ such that

$j = \operatorname{min} B_b$ and

$i \in B_a$ for

$a < b$ .
This is also the number of occurrences of the pattern {{1}, {2}}, such that 2 is the minimal 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.

Map

Yip

Description

A transformation of set partitions due to Yip.
Return the set partition of

$\{1,...,n\}$ corresponding to the set of arcs, interpreted as a rook placement, applying Yip's bijection

$\psi$ .