- FindStat

Identifier

Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => [3,2,2,2,1] => [3,2,2,2,1] => 5

[1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,8],[2,4,6,9,10]] => [3,2,3,2] => [3,3,2,2] => 4

[1,0,1,0,1,1,0,0,1,0] => [[1,3,5,6,9],[2,4,7,8,10]] => [3,3,3,1] => [3,3,3,1] => 4

[1,0,1,0,1,1,0,1,0,0] => [[1,3,5,6,8],[2,4,7,9,10]] => [3,3,2,2] => [3,3,2,2] => 4

[1,0,1,0,1,1,1,0,0,0] => [[1,3,5,6,7],[2,4,8,9,10]] => [3,4,3] => [4,3,3] => 3

[1,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,9],[2,5,6,8,10]] => [4,3,2,1] => [4,3,2,1] => 4

[1,0,1,1,0,0,1,1,0,0] => [[1,3,4,7,8],[2,5,6,9,10]] => [4,4,2] => [4,4,2] => 3

[1,0,1,1,0,1,0,0,1,0] => [[1,3,4,6,9],[2,5,7,8,10]] => [4,2,3,1] => [4,3,2,1] => 4

[1,0,1,1,0,1,0,1,0,0] => [[1,3,4,6,8],[2,5,7,9,10]] => [4,2,2,2] => [4,2,2,2] => 4

[1,0,1,1,0,1,1,0,0,0] => [[1,3,4,6,7],[2,5,8,9,10]] => [4,3,3] => [4,3,3] => 3

[1,0,1,1,1,0,0,0,1,0] => [[1,3,4,5,9],[2,6,7,8,10]] => [5,4,1] => [5,4,1] => 3

[1,0,1,1,1,0,0,1,0,0] => [[1,3,4,5,8],[2,6,7,9,10]] => [5,3,2] => [5,3,2] => 3

[1,0,1,1,1,0,1,0,0,0] => [[1,3,4,5,7],[2,6,8,9,10]] => [5,2,3] => [5,3,2] => 3

[1,0,1,1,1,1,0,0,0,0] => [[1,3,4,5,6],[2,7,8,9,10]] => [6,4] => [6,4] => 2

[1,1,0,0,1,0,1,0,1,0] => [[1,2,5,7,9],[3,4,6,8,10]] => [2,3,2,2,1] => [3,2,2,2,1] => 5

[1,1,0,0,1,0,1,1,0,0] => [[1,2,5,7,8],[3,4,6,9,10]] => [2,3,3,2] => [3,3,2,2] => 4

[1,1,0,0,1,1,0,0,1,0] => [[1,2,5,6,9],[3,4,7,8,10]] => [2,4,3,1] => [4,3,2,1] => 4

[1,1,0,0,1,1,0,1,0,0] => [[1,2,5,6,8],[3,4,7,9,10]] => [2,4,2,2] => [4,2,2,2] => 4

[1,1,0,0,1,1,1,0,0,0] => [[1,2,5,6,7],[3,4,8,9,10]] => [2,5,3] => [5,3,2] => 3

[1,1,0,1,0,0,1,0,1,0] => [[1,2,4,7,9],[3,5,6,8,10]] => [2,2,3,2,1] => [3,2,2,2,1] => 5

[1,1,0,1,0,0,1,1,0,0] => [[1,2,4,7,8],[3,5,6,9,10]] => [2,2,4,2] => [4,2,2,2] => 4

[1,1,0,1,0,1,0,0,1,0] => [[1,2,4,6,9],[3,5,7,8,10]] => [2,2,2,3,1] => [3,2,2,2,1] => 5

[1,1,0,1,0,1,0,1,0,0] => [[1,2,4,6,8],[3,5,7,9,10]] => [2,2,2,2,2] => [2,2,2,2,2] => 5

[1,1,0,1,0,1,1,0,0,0] => [[1,2,4,6,7],[3,5,8,9,10]] => [2,2,3,3] => [3,3,2,2] => 4

[1,1,0,1,1,0,0,0,1,0] => [[1,2,4,5,9],[3,6,7,8,10]] => [2,3,4,1] => [4,3,2,1] => 4

[1,1,0,1,1,0,0,1,0,0] => [[1,2,4,5,8],[3,6,7,9,10]] => [2,3,3,2] => [3,3,2,2] => 4

[1,1,0,1,1,0,1,0,0,0] => [[1,2,4,5,7],[3,6,8,9,10]] => [2,3,2,3] => [3,3,2,2] => 4

[1,1,0,1,1,1,0,0,0,0] => [[1,2,4,5,6],[3,7,8,9,10]] => [2,4,4] => [4,4,2] => 3

[1,1,1,0,0,0,1,0,1,0] => [[1,2,3,7,9],[4,5,6,8,10]] => [3,4,2,1] => [4,3,2,1] => 4

[1,1,1,0,0,0,1,1,0,0] => [[1,2,3,7,8],[4,5,6,9,10]] => [3,5,2] => [5,3,2] => 3

[1,1,1,0,0,1,0,0,1,0] => [[1,2,3,6,9],[4,5,7,8,10]] => [3,3,3,1] => [3,3,3,1] => 4

[1,1,1,0,0,1,0,1,0,0] => [[1,2,3,6,8],[4,5,7,9,10]] => [3,3,2,2] => [3,3,2,2] => 4

[1,1,1,0,0,1,1,0,0,0] => [[1,2,3,6,7],[4,5,8,9,10]] => [3,4,3] => [4,3,3] => 3

[1,1,1,0,1,0,0,0,1,0] => [[1,2,3,5,9],[4,6,7,8,10]] => [3,2,4,1] => [4,3,2,1] => 4

[1,1,1,0,1,0,0,1,0,0] => [[1,2,3,5,8],[4,6,7,9,10]] => [3,2,3,2] => [3,3,2,2] => 4

[1,1,1,0,1,0,1,0,0,0] => [[1,2,3,5,7],[4,6,8,9,10]] => [3,2,2,3] => [3,3,2,2] => 4

[1,1,1,0,1,1,0,0,0,0] => [[1,2,3,5,6],[4,7,8,9,10]] => [3,3,4] => [4,3,3] => 3

[1,1,1,1,0,0,0,0,1,0] => [[1,2,3,4,9],[5,6,7,8,10]] => [4,5,1] => [5,4,1] => 3

[1,1,1,1,0,0,0,1,0,0] => [[1,2,3,4,8],[5,6,7,9,10]] => [4,4,2] => [4,4,2] => 3

[1,1,1,1,0,0,1,0,0,0] => [[1,2,3,4,7],[5,6,8,9,10]] => [4,3,3] => [4,3,3] => 3

[1,1,1,1,0,1,0,0,0,0] => [[1,2,3,4,6],[5,7,8,9,10]] => [4,2,4] => [4,4,2] => 3

[1,1,1,1,1,0,0,0,0,0] => [[1,2,3,4,5],[6,7,8,9,10]] => [5,5] => [5,5] => 2

[1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => [3,2,2,2,2,1] => [3,2,2,2,2,1] => 6

[1,0,1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,9,10],[2,4,6,8,11,12]] => [3,2,2,3,2] => [3,3,2,2,2] => 5

[1,0,1,0,1,0,1,1,0,0,1,0] => [[1,3,5,7,8,11],[2,4,6,9,10,12]] => [3,2,3,3,1] => [3,3,3,2,1] => 5

[1,0,1,0,1,0,1,1,0,1,0,0] => [[1,3,5,7,8,10],[2,4,6,9,11,12]] => [3,2,3,2,2] => [3,3,2,2,2] => 5

[1,0,1,0,1,0,1,1,1,0,0,0] => [[1,3,5,7,8,9],[2,4,6,10,11,12]] => [3,2,4,3] => [4,3,3,2] => 4

[1,0,1,0,1,1,0,0,1,0,1,0] => [[1,3,5,6,9,11],[2,4,7,8,10,12]] => [3,3,3,2,1] => [3,3,3,2,1] => 5

[1,0,1,0,1,1,0,0,1,1,0,0] => [[1,3,5,6,9,10],[2,4,7,8,11,12]] => [3,3,4,2] => [4,3,3,2] => 4

[1,0,1,0,1,1,0,1,0,0,1,0] => [[1,3,5,6,8,11],[2,4,7,9,10,12]] => [3,3,2,3,1] => [3,3,3,2,1] => 5

[1,0,1,0,1,1,0,1,0,1,0,0] => [[1,3,5,6,8,10],[2,4,7,9,11,12]] => [3,3,2,2,2] => [3,3,2,2,2] => 5

[1,0,1,0,1,1,0,1,1,0,0,0] => [[1,3,5,6,8,9],[2,4,7,10,11,12]] => [3,3,3,3] => [3,3,3,3] => 4

[1,0,1,0,1,1,1,0,0,0,1,0] => [[1,3,5,6,7,11],[2,4,8,9,10,12]] => [3,4,4,1] => [4,4,3,1] => 4

[1,0,1,0,1,1,1,0,0,1,0,0] => [[1,3,5,6,7,10],[2,4,8,9,11,12]] => [3,4,3,2] => [4,3,3,2] => 4

[1,0,1,0,1,1,1,0,1,0,0,0] => [[1,3,5,6,7,9],[2,4,8,10,11,12]] => [3,4,2,3] => [4,3,3,2] => 4

[1,0,1,0,1,1,1,1,0,0,0,0] => [[1,3,5,6,7,8],[2,4,9,10,11,12]] => [3,5,4] => [5,4,3] => 3

[1,0,1,1,0,0,1,0,1,0,1,0] => [[1,3,4,7,9,11],[2,5,6,8,10,12]] => [4,3,2,2,1] => [4,3,2,2,1] => 5

[1,0,1,1,0,0,1,0,1,1,0,0] => [[1,3,4,7,9,10],[2,5,6,8,11,12]] => [4,3,3,2] => [4,3,3,2] => 4

[1,0,1,1,0,0,1,1,0,0,1,0] => [[1,3,4,7,8,11],[2,5,6,9,10,12]] => [4,4,3,1] => [4,4,3,1] => 4

[1,0,1,1,0,0,1,1,0,1,0,0] => [[1,3,4,7,8,10],[2,5,6,9,11,12]] => [4,4,2,2] => [4,4,2,2] => 4

[1,0,1,1,0,0,1,1,1,0,0,0] => [[1,3,4,7,8,9],[2,5,6,10,11,12]] => [4,5,3] => [5,4,3] => 3

[1,0,1,1,0,1,0,0,1,0,1,0] => [[1,3,4,6,9,11],[2,5,7,8,10,12]] => [4,2,3,2,1] => [4,3,2,2,1] => 5

[1,0,1,1,0,1,0,0,1,1,0,0] => [[1,3,4,6,9,10],[2,5,7,8,11,12]] => [4,2,4,2] => [4,4,2,2] => 4

[1,0,1,1,0,1,0,1,0,0,1,0] => [[1,3,4,6,8,11],[2,5,7,9,10,12]] => [4,2,2,3,1] => [4,3,2,2,1] => 5

[1,0,1,1,0,1,0,1,0,1,0,0] => [[1,3,4,6,8,10],[2,5,7,9,11,12]] => [4,2,2,2,2] => [4,2,2,2,2] => 5

[1,0,1,1,0,1,0,1,1,0,0,0] => [[1,3,4,6,8,9],[2,5,7,10,11,12]] => [4,2,3,3] => [4,3,3,2] => 4

[1,0,1,1,0,1,1,0,0,0,1,0] => [[1,3,4,6,7,11],[2,5,8,9,10,12]] => [4,3,4,1] => [4,4,3,1] => 4

[1,0,1,1,0,1,1,0,0,1,0,0] => [[1,3,4,6,7,10],[2,5,8,9,11,12]] => [4,3,3,2] => [4,3,3,2] => 4

[1,0,1,1,0,1,1,0,1,0,0,0] => [[1,3,4,6,7,9],[2,5,8,10,11,12]] => [4,3,2,3] => [4,3,3,2] => 4

[1,0,1,1,0,1,1,1,0,0,0,0] => [[1,3,4,6,7,8],[2,5,9,10,11,12]] => [4,4,4] => [4,4,4] => 3

[1,0,1,1,1,0,0,0,1,0,1,0] => [[1,3,4,5,9,11],[2,6,7,8,10,12]] => [5,4,2,1] => [5,4,2,1] => 4

[1,0,1,1,1,0,0,0,1,1,0,0] => [[1,3,4,5,9,10],[2,6,7,8,11,12]] => [5,5,2] => [5,5,2] => 3

[1,0,1,1,1,0,0,1,0,0,1,0] => [[1,3,4,5,8,11],[2,6,7,9,10,12]] => [5,3,3,1] => [5,3,3,1] => 4

[1,0,1,1,1,0,0,1,0,1,0,0] => [[1,3,4,5,8,10],[2,6,7,9,11,12]] => [5,3,2,2] => [5,3,2,2] => 4

[1,0,1,1,1,0,0,1,1,0,0,0] => [[1,3,4,5,8,9],[2,6,7,10,11,12]] => [5,4,3] => [5,4,3] => 3

[1,0,1,1,1,0,1,0,0,0,1,0] => [[1,3,4,5,7,11],[2,6,8,9,10,12]] => [5,2,4,1] => [5,4,2,1] => 4

[1,0,1,1,1,0,1,0,0,1,0,0] => [[1,3,4,5,7,10],[2,6,8,9,11,12]] => [5,2,3,2] => [5,3,2,2] => 4

[1,0,1,1,1,0,1,0,1,0,0,0] => [[1,3,4,5,7,9],[2,6,8,10,11,12]] => [5,2,2,3] => [5,3,2,2] => 4

[1,0,1,1,1,0,1,1,0,0,0,0] => [[1,3,4,5,7,8],[2,6,9,10,11,12]] => [5,3,4] => [5,4,3] => 3

>>> Load all 196 entries. <<<

[1,0,1,1,1,1,0,0,0,0,1,0] => [[1,3,4,5,6,11],[2,7,8,9,10,12]] => [6,5,1] => [6,5,1] => 3

[1,0,1,1,1,1,0,0,0,1,0,0] => [[1,3,4,5,6,10],[2,7,8,9,11,12]] => [6,4,2] => [6,4,2] => 3

[1,0,1,1,1,1,0,0,1,0,0,0] => [[1,3,4,5,6,9],[2,7,8,10,11,12]] => [6,3,3] => [6,3,3] => 3

[1,0,1,1,1,1,0,1,0,0,0,0] => [[1,3,4,5,6,8],[2,7,9,10,11,12]] => [6,2,4] => [6,4,2] => 3

[1,0,1,1,1,1,1,0,0,0,0,0] => [[1,3,4,5,6,7],[2,8,9,10,11,12]] => [7,5] => [7,5] => 2

[1,1,0,0,1,0,1,0,1,0,1,0] => [[1,2,5,7,9,11],[3,4,6,8,10,12]] => [2,3,2,2,2,1] => [3,2,2,2,2,1] => 6

[1,1,0,0,1,0,1,0,1,1,0,0] => [[1,2,5,7,9,10],[3,4,6,8,11,12]] => [2,3,2,3,2] => [3,3,2,2,2] => 5

[1,1,0,0,1,0,1,1,0,0,1,0] => [[1,2,5,7,8,11],[3,4,6,9,10,12]] => [2,3,3,3,1] => [3,3,3,2,1] => 5

[1,1,0,0,1,0,1,1,0,1,0,0] => [[1,2,5,7,8,10],[3,4,6,9,11,12]] => [2,3,3,2,2] => [3,3,2,2,2] => 5

[1,1,0,0,1,0,1,1,1,0,0,0] => [[1,2,5,7,8,9],[3,4,6,10,11,12]] => [2,3,4,3] => [4,3,3,2] => 4

[1,1,0,0,1,1,0,0,1,0,1,0] => [[1,2,5,6,9,11],[3,4,7,8,10,12]] => [2,4,3,2,1] => [4,3,2,2,1] => 5

[1,1,0,0,1,1,0,0,1,1,0,0] => [[1,2,5,6,9,10],[3,4,7,8,11,12]] => [2,4,4,2] => [4,4,2,2] => 4

[1,1,0,0,1,1,0,1,0,0,1,0] => [[1,2,5,6,8,11],[3,4,7,9,10,12]] => [2,4,2,3,1] => [4,3,2,2,1] => 5

[1,1,0,0,1,1,0,1,0,1,0,0] => [[1,2,5,6,8,10],[3,4,7,9,11,12]] => [2,4,2,2,2] => [4,2,2,2,2] => 5

[1,1,0,0,1,1,0,1,1,0,0,0] => [[1,2,5,6,8,9],[3,4,7,10,11,12]] => [2,4,3,3] => [4,3,3,2] => 4

[1,1,0,0,1,1,1,0,0,0,1,0] => [[1,2,5,6,7,11],[3,4,8,9,10,12]] => [2,5,4,1] => [5,4,2,1] => 4

[1,1,0,0,1,1,1,0,0,1,0,0] => [[1,2,5,6,7,10],[3,4,8,9,11,12]] => [2,5,3,2] => [5,3,2,2] => 4

[1,1,0,0,1,1,1,0,1,0,0,0] => [[1,2,5,6,7,9],[3,4,8,10,11,12]] => [2,5,2,3] => [5,3,2,2] => 4

[1,1,0,0,1,1,1,1,0,0,0,0] => [[1,2,5,6,7,8],[3,4,9,10,11,12]] => [2,6,4] => [6,4,2] => 3

[1,1,0,1,0,0,1,0,1,0,1,0] => [[1,2,4,7,9,11],[3,5,6,8,10,12]] => [2,2,3,2,2,1] => [3,2,2,2,2,1] => 6

[1,1,0,1,0,0,1,0,1,1,0,0] => [[1,2,4,7,9,10],[3,5,6,8,11,12]] => [2,2,3,3,2] => [3,3,2,2,2] => 5

[1,1,0,1,0,0,1,1,0,0,1,0] => [[1,2,4,7,8,11],[3,5,6,9,10,12]] => [2,2,4,3,1] => [4,3,2,2,1] => 5

[1,1,0,1,0,0,1,1,0,1,0,0] => [[1,2,4,7,8,10],[3,5,6,9,11,12]] => [2,2,4,2,2] => [4,2,2,2,2] => 5

[1,1,0,1,0,0,1,1,1,0,0,0] => [[1,2,4,7,8,9],[3,5,6,10,11,12]] => [2,2,5,3] => [5,3,2,2] => 4

[1,1,0,1,0,1,0,0,1,0,1,0] => [[1,2,4,6,9,11],[3,5,7,8,10,12]] => [2,2,2,3,2,1] => [3,2,2,2,2,1] => 6

[1,1,0,1,0,1,0,0,1,1,0,0] => [[1,2,4,6,9,10],[3,5,7,8,11,12]] => [2,2,2,4,2] => [4,2,2,2,2] => 5

[1,1,0,1,0,1,0,1,0,0,1,0] => [[1,2,4,6,8,11],[3,5,7,9,10,12]] => [2,2,2,2,3,1] => [3,2,2,2,2,1] => 6

[1,1,0,1,0,1,0,1,0,1,0,0] => [[1,2,4,6,8,10],[3,5,7,9,11,12]] => [2,2,2,2,2,2] => [2,2,2,2,2,2] => 6

[1,1,0,1,0,1,0,1,1,0,0,0] => [[1,2,4,6,8,9],[3,5,7,10,11,12]] => [2,2,2,3,3] => [3,3,2,2,2] => 5

[1,1,0,1,0,1,1,0,0,0,1,0] => [[1,2,4,6,7,11],[3,5,8,9,10,12]] => [2,2,3,4,1] => [4,3,2,2,1] => 5

[1,1,0,1,0,1,1,0,0,1,0,0] => [[1,2,4,6,7,10],[3,5,8,9,11,12]] => [2,2,3,3,2] => [3,3,2,2,2] => 5

[1,1,0,1,0,1,1,0,1,0,0,0] => [[1,2,4,6,7,9],[3,5,8,10,11,12]] => [2,2,3,2,3] => [3,3,2,2,2] => 5

[1,1,0,1,0,1,1,1,0,0,0,0] => [[1,2,4,6,7,8],[3,5,9,10,11,12]] => [2,2,4,4] => [4,4,2,2] => 4

[1,1,0,1,1,0,0,0,1,0,1,0] => [[1,2,4,5,9,11],[3,6,7,8,10,12]] => [2,3,4,2,1] => [4,3,2,2,1] => 5

[1,1,0,1,1,0,0,0,1,1,0,0] => [[1,2,4,5,9,10],[3,6,7,8,11,12]] => [2,3,5,2] => [5,3,2,2] => 4

[1,1,0,1,1,0,0,1,0,0,1,0] => [[1,2,4,5,8,11],[3,6,7,9,10,12]] => [2,3,3,3,1] => [3,3,3,2,1] => 5

[1,1,0,1,1,0,0,1,0,1,0,0] => [[1,2,4,5,8,10],[3,6,7,9,11,12]] => [2,3,3,2,2] => [3,3,2,2,2] => 5

[1,1,0,1,1,0,0,1,1,0,0,0] => [[1,2,4,5,8,9],[3,6,7,10,11,12]] => [2,3,4,3] => [4,3,3,2] => 4

[1,1,0,1,1,0,1,0,0,0,1,0] => [[1,2,4,5,7,11],[3,6,8,9,10,12]] => [2,3,2,4,1] => [4,3,2,2,1] => 5

[1,1,0,1,1,0,1,0,0,1,0,0] => [[1,2,4,5,7,10],[3,6,8,9,11,12]] => [2,3,2,3,2] => [3,3,2,2,2] => 5

[1,1,0,1,1,0,1,0,1,0,0,0] => [[1,2,4,5,7,9],[3,6,8,10,11,12]] => [2,3,2,2,3] => [3,3,2,2,2] => 5

[1,1,0,1,1,0,1,1,0,0,0,0] => [[1,2,4,5,7,8],[3,6,9,10,11,12]] => [2,3,3,4] => [4,3,3,2] => 4

[1,1,0,1,1,1,0,0,0,0,1,0] => [[1,2,4,5,6,11],[3,7,8,9,10,12]] => [2,4,5,1] => [5,4,2,1] => 4

[1,1,0,1,1,1,0,0,0,1,0,0] => [[1,2,4,5,6,10],[3,7,8,9,11,12]] => [2,4,4,2] => [4,4,2,2] => 4

[1,1,0,1,1,1,0,0,1,0,0,0] => [[1,2,4,5,6,9],[3,7,8,10,11,12]] => [2,4,3,3] => [4,3,3,2] => 4

[1,1,0,1,1,1,0,1,0,0,0,0] => [[1,2,4,5,6,8],[3,7,9,10,11,12]] => [2,4,2,4] => [4,4,2,2] => 4

[1,1,0,1,1,1,1,0,0,0,0,0] => [[1,2,4,5,6,7],[3,8,9,10,11,12]] => [2,5,5] => [5,5,2] => 3

[1,1,1,0,0,0,1,0,1,0,1,0] => [[1,2,3,7,9,11],[4,5,6,8,10,12]] => [3,4,2,2,1] => [4,3,2,2,1] => 5

[1,1,1,0,0,0,1,0,1,1,0,0] => [[1,2,3,7,9,10],[4,5,6,8,11,12]] => [3,4,3,2] => [4,3,3,2] => 4

[1,1,1,0,0,0,1,1,0,0,1,0] => [[1,2,3,7,8,11],[4,5,6,9,10,12]] => [3,5,3,1] => [5,3,3,1] => 4

[1,1,1,0,0,0,1,1,0,1,0,0] => [[1,2,3,7,8,10],[4,5,6,9,11,12]] => [3,5,2,2] => [5,3,2,2] => 4

[1,1,1,0,0,0,1,1,1,0,0,0] => [[1,2,3,7,8,9],[4,5,6,10,11,12]] => [3,6,3] => [6,3,3] => 3

[1,1,1,0,0,1,0,0,1,0,1,0] => [[1,2,3,6,9,11],[4,5,7,8,10,12]] => [3,3,3,2,1] => [3,3,3,2,1] => 5

[1,1,1,0,0,1,0,0,1,1,0,0] => [[1,2,3,6,9,10],[4,5,7,8,11,12]] => [3,3,4,2] => [4,3,3,2] => 4

[1,1,1,0,0,1,0,1,0,0,1,0] => [[1,2,3,6,8,11],[4,5,7,9,10,12]] => [3,3,2,3,1] => [3,3,3,2,1] => 5

[1,1,1,0,0,1,0,1,0,1,0,0] => [[1,2,3,6,8,10],[4,5,7,9,11,12]] => [3,3,2,2,2] => [3,3,2,2,2] => 5

[1,1,1,0,0,1,0,1,1,0,0,0] => [[1,2,3,6,8,9],[4,5,7,10,11,12]] => [3,3,3,3] => [3,3,3,3] => 4

[1,1,1,0,0,1,1,0,0,0,1,0] => [[1,2,3,6,7,11],[4,5,8,9,10,12]] => [3,4,4,1] => [4,4,3,1] => 4

[1,1,1,0,0,1,1,0,0,1,0,0] => [[1,2,3,6,7,10],[4,5,8,9,11,12]] => [3,4,3,2] => [4,3,3,2] => 4

[1,1,1,0,0,1,1,0,1,0,0,0] => [[1,2,3,6,7,9],[4,5,8,10,11,12]] => [3,4,2,3] => [4,3,3,2] => 4

[1,1,1,0,0,1,1,1,0,0,0,0] => [[1,2,3,6,7,8],[4,5,9,10,11,12]] => [3,5,4] => [5,4,3] => 3

[1,1,1,0,1,0,0,0,1,0,1,0] => [[1,2,3,5,9,11],[4,6,7,8,10,12]] => [3,2,4,2,1] => [4,3,2,2,1] => 5

[1,1,1,0,1,0,0,0,1,1,0,0] => [[1,2,3,5,9,10],[4,6,7,8,11,12]] => [3,2,5,2] => [5,3,2,2] => 4

[1,1,1,0,1,0,0,1,0,0,1,0] => [[1,2,3,5,8,11],[4,6,7,9,10,12]] => [3,2,3,3,1] => [3,3,3,2,1] => 5

[1,1,1,0,1,0,0,1,0,1,0,0] => [[1,2,3,5,8,10],[4,6,7,9,11,12]] => [3,2,3,2,2] => [3,3,2,2,2] => 5

[1,1,1,0,1,0,0,1,1,0,0,0] => [[1,2,3,5,8,9],[4,6,7,10,11,12]] => [3,2,4,3] => [4,3,3,2] => 4

[1,1,1,0,1,0,1,0,0,0,1,0] => [[1,2,3,5,7,11],[4,6,8,9,10,12]] => [3,2,2,4,1] => [4,3,2,2,1] => 5

[1,1,1,0,1,0,1,0,0,1,0,0] => [[1,2,3,5,7,10],[4,6,8,9,11,12]] => [3,2,2,3,2] => [3,3,2,2,2] => 5

[1,1,1,0,1,0,1,0,1,0,0,0] => [[1,2,3,5,7,9],[4,6,8,10,11,12]] => [3,2,2,2,3] => [3,3,2,2,2] => 5

[1,1,1,0,1,0,1,1,0,0,0,0] => [[1,2,3,5,7,8],[4,6,9,10,11,12]] => [3,2,3,4] => [4,3,3,2] => 4

[1,1,1,0,1,1,0,0,0,0,1,0] => [[1,2,3,5,6,11],[4,7,8,9,10,12]] => [3,3,5,1] => [5,3,3,1] => 4

[1,1,1,0,1,1,0,0,0,1,0,0] => [[1,2,3,5,6,10],[4,7,8,9,11,12]] => [3,3,4,2] => [4,3,3,2] => 4

[1,1,1,0,1,1,0,0,1,0,0,0] => [[1,2,3,5,6,9],[4,7,8,10,11,12]] => [3,3,3,3] => [3,3,3,3] => 4

[1,1,1,0,1,1,0,1,0,0,0,0] => [[1,2,3,5,6,8],[4,7,9,10,11,12]] => [3,3,2,4] => [4,3,3,2] => 4

[1,1,1,0,1,1,1,0,0,0,0,0] => [[1,2,3,5,6,7],[4,8,9,10,11,12]] => [3,4,5] => [5,4,3] => 3

[1,1,1,1,0,0,0,0,1,0,1,0] => [[1,2,3,4,9,11],[5,6,7,8,10,12]] => [4,5,2,1] => [5,4,2,1] => 4

[1,1,1,1,0,0,0,0,1,1,0,0] => [[1,2,3,4,9,10],[5,6,7,8,11,12]] => [4,6,2] => [6,4,2] => 3

[1,1,1,1,0,0,0,1,0,0,1,0] => [[1,2,3,4,8,11],[5,6,7,9,10,12]] => [4,4,3,1] => [4,4,3,1] => 4

[1,1,1,1,0,0,0,1,0,1,0,0] => [[1,2,3,4,8,10],[5,6,7,9,11,12]] => [4,4,2,2] => [4,4,2,2] => 4

[1,1,1,1,0,0,0,1,1,0,0,0] => [[1,2,3,4,8,9],[5,6,7,10,11,12]] => [4,5,3] => [5,4,3] => 3

[1,1,1,1,0,0,1,0,0,0,1,0] => [[1,2,3,4,7,11],[5,6,8,9,10,12]] => [4,3,4,1] => [4,4,3,1] => 4

[1,1,1,1,0,0,1,0,0,1,0,0] => [[1,2,3,4,7,10],[5,6,8,9,11,12]] => [4,3,3,2] => [4,3,3,2] => 4

[1,1,1,1,0,0,1,0,1,0,0,0] => [[1,2,3,4,7,9],[5,6,8,10,11,12]] => [4,3,2,3] => [4,3,3,2] => 4

[1,1,1,1,0,0,1,1,0,0,0,0] => [[1,2,3,4,7,8],[5,6,9,10,11,12]] => [4,4,4] => [4,4,4] => 3

[1,1,1,1,0,1,0,0,0,0,1,0] => [[1,2,3,4,6,11],[5,7,8,9,10,12]] => [4,2,5,1] => [5,4,2,1] => 4

[1,1,1,1,0,1,0,0,0,1,0,0] => [[1,2,3,4,6,10],[5,7,8,9,11,12]] => [4,2,4,2] => [4,4,2,2] => 4

[1,1,1,1,0,1,0,0,1,0,0,0] => [[1,2,3,4,6,9],[5,7,8,10,11,12]] => [4,2,3,3] => [4,3,3,2] => 4

[1,1,1,1,0,1,0,1,0,0,0,0] => [[1,2,3,4,6,8],[5,7,9,10,11,12]] => [4,2,2,4] => [4,4,2,2] => 4

[1,1,1,1,0,1,1,0,0,0,0,0] => [[1,2,3,4,6,7],[5,8,9,10,11,12]] => [4,3,5] => [5,4,3] => 3

[1,1,1,1,1,0,0,0,0,0,1,0] => [[1,2,3,4,5,11],[6,7,8,9,10,12]] => [5,6,1] => [6,5,1] => 3

[1,1,1,1,1,0,0,0,0,1,0,0] => [[1,2,3,4,5,10],[6,7,8,9,11,12]] => [5,5,2] => [5,5,2] => 3

[1,1,1,1,1,0,0,0,1,0,0,0] => [[1,2,3,4,5,9],[6,7,8,10,11,12]] => [5,4,3] => [5,4,3] => 3

[1,1,1,1,1,0,0,1,0,0,0,0] => [[1,2,3,4,5,8],[6,7,9,10,11,12]] => [5,3,4] => [5,4,3] => 3

[1,1,1,1,1,0,1,0,0,0,0,0] => [[1,2,3,4,5,7],[6,8,9,10,11,12]] => [5,2,5] => [5,5,2] => 3

[1,1,1,1,1,1,0,0,0,0,0,0] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => [6,6] => [6,6] => 2

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$F_{1} = q$

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

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

$F_{4} = 2\ q^{2} + 8\ q^{3} + 4\ q^{4}$

$F_{5} = 2\ q^{2} + 15\ q^{3} + 20\ q^{4} + 5\ q^{5}$

$F_{6} = 2\ q^{2} + 24\ q^{3} + 60\ q^{4} + 40\ q^{5} + 6\ q^{6}$

Description

The length of the partition.

Map

peak composition

Description

The composition corresponding to the peak set of a standard tableau.
Let

$T$ be a standard tableau of size

$n$ .
An entry

$i$ of

$T$ is a descent if

$i+1$ is in a lower row (in English notation), otherwise

$i$ is an ascent.
An entry

$2 \leq i \leq n-1$ is a peak, if

$i-1$ is an ascent and

$i$ is a descent.
This map returns the composition

$c_1,\dots,c_k$ of

$n$ such that

$\{c_1, c_1+c_2,\dots, c_1+\dots+c_k\}$ is the peak set of

$T$ .

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.

Map

to two-row standard tableau

Description

Return a standard tableau of shape

$(n,n)$ where

$n$ is the semilength of the Dyck path.
Given a Dyck path

$D$ , its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.