- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001604: Integer partitions ⟶ ℤ

Values

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

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

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

[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => [[4,4,4,4,4],[3]] => [3] => 1

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

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

[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => [[5,5,5,5,5],[4]] => [4] => 1

[7,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => [[5,5,5,5],[4]] => [4] => 1

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

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

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

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

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

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

[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => [[5,5,5,5,5,5],[4]] => [4] => 1

[8,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0] => [[4,4,4,4,4,4],[3]] => [3] => 1

[7,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0] => [[4,4,4,4,4],[3,1]] => [3,1] => 0

[7,1,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0] => [[4,4,4,4,3],[3]] => [3] => 1

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

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

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

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

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

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

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

[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [[6,6,6,6,6,6],[5]] => [5] => 1

[9,1] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0] => [[6,6,6,6,6],[5]] => [5] => 1

[8,2] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0] => [[5,5,5,5,4],[4]] => [4] => 1

[8,1,1] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0] => [[5,5,5,5,5],[4,1]] => [4,1] => 0

[7,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => [[5,5,5,4],[4]] => [4] => 1

[7,1,1,1] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0] => [[5,5,5,5],[4,1]] => [4,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[7,4,1,1] => [1,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0] => [[4,4,4,4,4],[3,2,1]] => [3,2,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[7,6,1,1] => [1,1,1,1,0,1,1,0,0,0,0,0,1,0,1,0] => [[4,4,4,4,4],[3,3,1]] => [3,3,1] => 0

[7,4,3,1] => [1,1,1,1,0,1,0,0,1,0,1,0,0,0,1,0] => [[4,4,4,4,4],[3,2,2]] => [3,2,2] => 3

[7,4,1,1,1,1] => [1,1,0,1,1,1,1,0,0,0,1,0,0,0,1,0] => [[4,4,4,4,4],[3,2,1,1]] => [3,2,1,1] => 2

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

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

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

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

[7,2,2,2,2,1] => [1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0] => [[5,5,5,5],[4,2,2]] => [4,2,2] => 7

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

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

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

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

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

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

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

[7,4,3,1,1,1] => [1,1,0,1,1,1,0,0,1,0,1,0,0,0,1,0] => [[4,4,4,4,4],[3,2,2,1]] => [3,2,2,1] => 4

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

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

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

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

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

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

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

>>> Load all 166 entries. <<<

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[7,6,5,4,2,1] => [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0] => [[4,4,4,4,4],[3,3,3,3]] => [3,3,3,3] => 26

[7,6,5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0] => [[5,5,5,5],[4,4,4]] => [4,4,4] => 32

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

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

[7,6,5,2,2,1] => [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0] => [[4,4,4,4,4],[3,3,3,2]] => [3,3,3,2] => 16

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

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

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

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

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

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

[8,7,6,5,3,2,1] => [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0] => [[5,5,5,5,5],[4,4,4,4]] => [4,4,4,4] => 831

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

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

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

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

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

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

search for individual values

searching the database for the individual values of this statistic

Description

The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

Map

skew partition

Description

The parallelogram polyomino corresponding to a Dyck path, interpreted as a skew partition.
Let

$D$ be a Dyck path of semilength

$n$ . The parallelogram polyomino

$\gamma(D)$ is defined as follows: let

$\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to

$D$ . Then, the upper path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with even indices, and the lower path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with odd indices.
This map returns the skew partition definded by the diagram of

$\gamma(D)$ .

Map

inner shape

Description

The inner shape of a skew partition.

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.