- FindStat

Identifier

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000504: Set partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 205 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$F_{5} = 14\ q^{2} + 14\ q^{3} + 9\ q^{4} + 4\ q^{5} + q^{6}$

$F_{6} = 42\ q^{2} + 42\ q^{3} + 28\ q^{4} + 14\ q^{5} + 5\ q^{6} + q^{7}$

Description

The cardinality of the first block of a set partition.
The number of partitions of

$\{1,\ldots,n\}$ into

$k$ blocks in which the first block has cardinality

$j+1$ is given by

$\binom{n-1}{j}S(n-j-1,k-1)$ , see [1, Theorem 1.1] and the references therein. Here,

$S(n,k)$ are the Stirling numbers of the second kind counting all set partitions of

$\{1,\ldots,n\}$ into

$k$ blocks [2].

Map

inverse

Description

Sends a permutation to its inverse.

Map

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.

Map

weak exceedance partition

Description

The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs

$(i, \pi(i))$ with

$i\leq\pi(i)$ .