- FindStat

Identifier

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St001062: Set partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[7] => [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}} => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 163 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Description

The maximal size of a block of a set partition.

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

Map

to 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).

Map

to cycle type

Description

Let

$\pi=c_1\dots c_r$ a permutation of size

$n$ decomposed in its cyclic parts. The associated set partition of

$[n]$ then is

$S=S_1\cup\dots\cup S_r$ such that

$S_i$ is the set of integers in the cycle

$c_i$ .
A permutation is cyclic [1] if and only if its cycle type is a hook partition [2].