Identifier
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00181: Skew partitions —row lengths⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Images
[1,0] => [[1],[]] => [1] => [1]
[1,0,1,0] => [[1,1],[]] => [1,1] => [1,1]
[1,1,0,0] => [[2],[]] => [2] => [2]
[1,0,1,0,1,0] => [[1,1,1],[]] => [1,1,1] => [1,1,1]
[1,0,1,1,0,0] => [[2,1],[]] => [2,1] => [1,2]
[1,1,0,0,1,0] => [[2,2],[1]] => [1,2] => [2,1]
[1,1,0,1,0,0] => [[3],[]] => [3] => [3]
[1,1,1,0,0,0] => [[2,2],[]] => [2,2] => [2,2]
[1,0,1,0,1,0,1,0] => [[1,1,1,1],[]] => [1,1,1,1] => [1,1,1,1]
[1,0,1,0,1,1,0,0] => [[2,1,1],[]] => [2,1,1] => [1,1,2]
[1,0,1,1,0,0,1,0] => [[2,2,1],[1]] => [1,2,1] => [1,2,1]
[1,0,1,1,0,1,0,0] => [[3,1],[]] => [3,1] => [1,3]
[1,0,1,1,1,0,0,0] => [[2,2,1],[]] => [2,2,1] => [1,2,2]
[1,1,0,0,1,0,1,0] => [[2,2,2],[1,1]] => [1,1,2] => [2,1,1]
[1,1,0,0,1,1,0,0] => [[3,2],[1]] => [2,2] => [2,2]
[1,1,0,1,0,0,1,0] => [[3,3],[2]] => [1,3] => [3,1]
[1,1,0,1,0,1,0,0] => [[4],[]] => [4] => [4]
[1,1,0,1,1,0,0,0] => [[3,3],[1]] => [2,3] => [3,2]
[1,1,1,0,0,0,1,0] => [[2,2,2],[1]] => [1,2,2] => [2,2,1]
[1,1,1,0,0,1,0,0] => [[3,2],[]] => [3,2] => [2,3]
[1,1,1,0,1,0,0,0] => [[2,2,2],[]] => [2,2,2] => [2,2,2]
[1,1,1,1,0,0,0,0] => [[3,3],[]] => [3,3] => [3,3]
[1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]] => [1,1,1,1,1] => [1,1,1,1,1]
[1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]] => [2,1,1,1] => [1,1,1,2]
[1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1],[1]] => [1,2,1,1] => [1,1,2,1]
[1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]] => [3,1,1] => [1,1,3]
[1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1],[]] => [2,2,1,1] => [1,1,2,2]
[1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1],[1,1]] => [1,1,2,1] => [1,2,1,1]
[1,0,1,1,0,0,1,1,0,0] => [[3,2,1],[1]] => [2,2,1] => [1,2,2]
[1,0,1,1,0,1,0,0,1,0] => [[3,3,1],[2]] => [1,3,1] => [1,3,1]
[1,0,1,1,0,1,0,1,0,0] => [[4,1],[]] => [4,1] => [1,4]
[1,0,1,1,0,1,1,0,0,0] => [[3,3,1],[1]] => [2,3,1] => [1,3,2]
[1,0,1,1,1,0,0,0,1,0] => [[2,2,2,1],[1]] => [1,2,2,1] => [1,2,2,1]
[1,0,1,1,1,0,0,1,0,0] => [[3,2,1],[]] => [3,2,1] => [1,2,3]
[1,0,1,1,1,0,1,0,0,0] => [[2,2,2,1],[]] => [2,2,2,1] => [1,2,2,2]
[1,0,1,1,1,1,0,0,0,0] => [[3,3,1],[]] => [3,3,1] => [1,3,3]
[1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2],[1,1,1]] => [1,1,1,2] => [2,1,1,1]
[1,1,0,0,1,0,1,1,0,0] => [[3,2,2],[1,1]] => [2,1,2] => [2,1,2]
[1,1,0,0,1,1,0,0,1,0] => [[3,3,2],[2,1]] => [1,2,2] => [2,2,1]
[1,1,0,0,1,1,0,1,0,0] => [[4,2],[1]] => [3,2] => [2,3]
[1,1,0,0,1,1,1,0,0,0] => [[3,3,2],[1,1]] => [2,2,2] => [2,2,2]
[1,1,0,1,0,0,1,0,1,0] => [[3,3,3],[2,2]] => [1,1,3] => [3,1,1]
[1,1,0,1,0,0,1,1,0,0] => [[4,3],[2]] => [2,3] => [3,2]
[1,1,0,1,0,1,0,0,1,0] => [[4,4],[3]] => [1,4] => [4,1]
[1,1,0,1,0,1,0,1,0,0] => [[5],[]] => [5] => [5]
[1,1,0,1,0,1,1,0,0,0] => [[4,4],[2]] => [2,4] => [4,2]
[1,1,0,1,1,0,0,0,1,0] => [[3,3,3],[2,1]] => [1,2,3] => [3,2,1]
[1,1,0,1,1,0,0,1,0,0] => [[4,3],[1]] => [3,3] => [3,3]
[1,1,0,1,1,0,1,0,0,0] => [[3,3,3],[1,1]] => [2,2,3] => [3,2,2]
[1,1,0,1,1,1,0,0,0,0] => [[4,4],[1]] => [3,4] => [4,3]
[1,1,1,0,0,0,1,0,1,0] => [[2,2,2,2],[1,1]] => [1,1,2,2] => [2,2,1,1]
[1,1,1,0,0,0,1,1,0,0] => [[3,2,2],[1]] => [2,2,2] => [2,2,2]
[1,1,1,0,0,1,0,0,1,0] => [[3,3,2],[2]] => [1,3,2] => [2,3,1]
[1,1,1,0,0,1,0,1,0,0] => [[4,2],[]] => [4,2] => [2,4]
[1,1,1,0,0,1,1,0,0,0] => [[3,3,2],[1]] => [2,3,2] => [2,3,2]
[1,1,1,0,1,0,0,0,1,0] => [[2,2,2,2],[1]] => [1,2,2,2] => [2,2,2,1]
[1,1,1,0,1,0,0,1,0,0] => [[3,2,2],[]] => [3,2,2] => [2,2,3]
[1,1,1,0,1,0,1,0,0,0] => [[2,2,2,2],[]] => [2,2,2,2] => [2,2,2,2]
[1,1,1,0,1,1,0,0,0,0] => [[3,3,2],[]] => [3,3,2] => [2,3,3]
[1,1,1,1,0,0,0,0,1,0] => [[3,3,3],[2]] => [1,3,3] => [3,3,1]
[1,1,1,1,0,0,0,1,0,0] => [[4,3],[]] => [4,3] => [3,4]
[1,1,1,1,0,0,1,0,0,0] => [[3,3,3],[1]] => [2,3,3] => [3,3,2]
[1,1,1,1,0,1,0,0,0,0] => [[4,4],[]] => [4,4] => [4,4]
[1,1,1,1,1,0,0,0,0,0] => [[3,3,3],[]] => [3,3,3] => [3,3,3]
[1,0,1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1,1],[]] => [1,1,1,1,1,1] => [1,1,1,1,1,1]
[1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1],[]] => [2,1,1,1,1] => [1,1,1,1,2]
[1,0,1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1,1],[1]] => [1,2,1,1,1] => [1,1,1,2,1]
[1,0,1,0,1,0,1,1,0,1,0,0] => [[3,1,1,1],[]] => [3,1,1,1] => [1,1,1,3]
[1,0,1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1,1],[]] => [2,2,1,1,1] => [1,1,1,2,2]
[1,0,1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1,1],[1,1]] => [1,1,2,1,1] => [1,1,2,1,1]
[1,0,1,0,1,1,0,0,1,1,0,0] => [[3,2,1,1],[1]] => [2,2,1,1] => [1,1,2,2]
[1,0,1,0,1,1,0,1,0,0,1,0] => [[3,3,1,1],[2]] => [1,3,1,1] => [1,1,3,1]
[1,0,1,0,1,1,0,1,0,1,0,0] => [[4,1,1],[]] => [4,1,1] => [1,1,4]
[1,0,1,0,1,1,0,1,1,0,0,0] => [[3,3,1,1],[1]] => [2,3,1,1] => [1,1,3,2]
[1,0,1,0,1,1,1,0,0,0,1,0] => [[2,2,2,1,1],[1]] => [1,2,2,1,1] => [1,1,2,2,1]
[1,0,1,0,1,1,1,0,0,1,0,0] => [[3,2,1,1],[]] => [3,2,1,1] => [1,1,2,3]
[1,0,1,0,1,1,1,0,1,0,0,0] => [[2,2,2,1,1],[]] => [2,2,2,1,1] => [1,1,2,2,2]
[1,0,1,0,1,1,1,1,0,0,0,0] => [[3,3,1,1],[]] => [3,3,1,1] => [1,1,3,3]
[1,0,1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2,1],[1,1,1]] => [1,1,1,2,1] => [1,2,1,1,1]
[1,0,1,1,0,0,1,0,1,1,0,0] => [[3,2,2,1],[1,1]] => [2,1,2,1] => [1,2,1,2]
[1,0,1,1,0,0,1,1,0,0,1,0] => [[3,3,2,1],[2,1]] => [1,2,2,1] => [1,2,2,1]
[1,0,1,1,0,0,1,1,0,1,0,0] => [[4,2,1],[1]] => [3,2,1] => [1,2,3]
[1,0,1,1,0,0,1,1,1,0,0,0] => [[3,3,2,1],[1,1]] => [2,2,2,1] => [1,2,2,2]
[1,0,1,1,0,1,0,0,1,0,1,0] => [[3,3,3,1],[2,2]] => [1,1,3,1] => [1,3,1,1]
[1,0,1,1,0,1,0,0,1,1,0,0] => [[4,3,1],[2]] => [2,3,1] => [1,3,2]
[1,0,1,1,0,1,0,1,0,0,1,0] => [[4,4,1],[3]] => [1,4,1] => [1,4,1]
[1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1],[]] => [5,1] => [1,5]
[1,0,1,1,0,1,0,1,1,0,0,0] => [[4,4,1],[2]] => [2,4,1] => [1,4,2]
[1,0,1,1,0,1,1,0,0,0,1,0] => [[3,3,3,1],[2,1]] => [1,2,3,1] => [1,3,2,1]
[1,0,1,1,0,1,1,0,0,1,0,0] => [[4,3,1],[1]] => [3,3,1] => [1,3,3]
[1,0,1,1,0,1,1,0,1,0,0,0] => [[3,3,3,1],[1,1]] => [2,2,3,1] => [1,3,2,2]
[1,0,1,1,0,1,1,1,0,0,0,0] => [[4,4,1],[1]] => [3,4,1] => [1,4,3]
[1,0,1,1,1,0,0,0,1,0,1,0] => [[2,2,2,2,1],[1,1]] => [1,1,2,2,1] => [1,2,2,1,1]
[1,0,1,1,1,0,0,0,1,1,0,0] => [[3,2,2,1],[1]] => [2,2,2,1] => [1,2,2,2]
[1,0,1,1,1,0,0,1,0,0,1,0] => [[3,3,2,1],[2]] => [1,3,2,1] => [1,2,3,1]
[1,0,1,1,1,0,0,1,0,1,0,0] => [[4,2,1],[]] => [4,2,1] => [1,2,4]
[1,0,1,1,1,0,0,1,1,0,0,0] => [[3,3,2,1],[1]] => [2,3,2,1] => [1,2,3,2]
[1,0,1,1,1,0,1,0,0,0,1,0] => [[2,2,2,2,1],[1]] => [1,2,2,2,1] => [1,2,2,2,1]
[1,0,1,1,1,0,1,0,0,1,0,0] => [[3,2,2,1],[]] => [3,2,2,1] => [1,2,2,3]
[1,0,1,1,1,0,1,0,1,0,0,0] => [[2,2,2,2,1],[]] => [2,2,2,2,1] => [1,2,2,2,2]
[1,0,1,1,1,0,1,1,0,0,0,0] => [[3,3,2,1],[]] => [3,3,2,1] => [1,2,3,3]
>>> Load all 416 entries. <<<
[1,0,1,1,1,1,0,0,0,0,1,0] => [[3,3,3,1],[2]] => [1,3,3,1] => [1,3,3,1]
[1,0,1,1,1,1,0,0,0,1,0,0] => [[4,3,1],[]] => [4,3,1] => [1,3,4]
[1,0,1,1,1,1,0,0,1,0,0,0] => [[3,3,3,1],[1]] => [2,3,3,1] => [1,3,3,2]
[1,0,1,1,1,1,0,1,0,0,0,0] => [[4,4,1],[]] => [4,4,1] => [1,4,4]
[1,0,1,1,1,1,1,0,0,0,0,0] => [[3,3,3,1],[]] => [3,3,3,1] => [1,3,3,3]
[1,1,0,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1,2] => [2,1,1,1,1]
[1,1,0,0,1,0,1,0,1,1,0,0] => [[3,2,2,2],[1,1,1]] => [2,1,1,2] => [2,1,1,2]
[1,1,0,0,1,0,1,1,0,0,1,0] => [[3,3,2,2],[2,1,1]] => [1,2,1,2] => [2,1,2,1]
[1,1,0,0,1,0,1,1,0,1,0,0] => [[4,2,2],[1,1]] => [3,1,2] => [2,1,3]
[1,1,0,0,1,0,1,1,1,0,0,0] => [[3,3,2,2],[1,1,1]] => [2,2,1,2] => [2,1,2,2]
[1,1,0,0,1,1,0,0,1,0,1,0] => [[3,3,3,2],[2,2,1]] => [1,1,2,2] => [2,2,1,1]
[1,1,0,0,1,1,0,0,1,1,0,0] => [[4,3,2],[2,1]] => [2,2,2] => [2,2,2]
[1,1,0,0,1,1,0,1,0,0,1,0] => [[4,4,2],[3,1]] => [1,3,2] => [2,3,1]
[1,1,0,0,1,1,0,1,0,1,0,0] => [[5,2],[1]] => [4,2] => [2,4]
[1,1,0,0,1,1,0,1,1,0,0,0] => [[4,4,2],[2,1]] => [2,3,2] => [2,3,2]
[1,1,0,0,1,1,1,0,0,0,1,0] => [[3,3,3,2],[2,1,1]] => [1,2,2,2] => [2,2,2,1]
[1,1,0,0,1,1,1,0,0,1,0,0] => [[4,3,2],[1,1]] => [3,2,2] => [2,2,3]
[1,1,0,0,1,1,1,0,1,0,0,0] => [[3,3,3,2],[1,1,1]] => [2,2,2,2] => [2,2,2,2]
[1,1,0,0,1,1,1,1,0,0,0,0] => [[4,4,2],[1,1]] => [3,3,2] => [2,3,3]
[1,1,0,1,0,0,1,0,1,0,1,0] => [[3,3,3,3],[2,2,2]] => [1,1,1,3] => [3,1,1,1]
[1,1,0,1,0,0,1,0,1,1,0,0] => [[4,3,3],[2,2]] => [2,1,3] => [3,1,2]
[1,1,0,1,0,0,1,1,0,0,1,0] => [[4,4,3],[3,2]] => [1,2,3] => [3,2,1]
[1,1,0,1,0,0,1,1,0,1,0,0] => [[5,3],[2]] => [3,3] => [3,3]
[1,1,0,1,0,0,1,1,1,0,0,0] => [[4,4,3],[2,2]] => [2,2,3] => [3,2,2]
[1,1,0,1,0,1,0,0,1,0,1,0] => [[4,4,4],[3,3]] => [1,1,4] => [4,1,1]
[1,1,0,1,0,1,0,0,1,1,0,0] => [[5,4],[3]] => [2,4] => [4,2]
[1,1,0,1,0,1,0,1,0,0,1,0] => [[5,5],[4]] => [1,5] => [5,1]
[1,1,0,1,0,1,0,1,0,1,0,0] => [[6],[]] => [6] => [6]
[1,1,0,1,0,1,0,1,1,0,0,0] => [[5,5],[3]] => [2,5] => [5,2]
[1,1,0,1,0,1,1,0,0,0,1,0] => [[4,4,4],[3,2]] => [1,2,4] => [4,2,1]
[1,1,0,1,0,1,1,0,0,1,0,0] => [[5,4],[2]] => [3,4] => [4,3]
[1,1,0,1,0,1,1,0,1,0,0,0] => [[4,4,4],[2,2]] => [2,2,4] => [4,2,2]
[1,1,0,1,0,1,1,1,0,0,0,0] => [[5,5],[2]] => [3,5] => [5,3]
[1,1,0,1,1,0,0,0,1,0,1,0] => [[3,3,3,3],[2,2,1]] => [1,1,2,3] => [3,2,1,1]
[1,1,0,1,1,0,0,0,1,1,0,0] => [[4,3,3],[2,1]] => [2,2,3] => [3,2,2]
[1,1,0,1,1,0,0,1,0,0,1,0] => [[4,4,3],[3,1]] => [1,3,3] => [3,3,1]
[1,1,0,1,1,0,0,1,0,1,0,0] => [[5,3],[1]] => [4,3] => [3,4]
[1,1,0,1,1,0,0,1,1,0,0,0] => [[4,4,3],[2,1]] => [2,3,3] => [3,3,2]
[1,1,0,1,1,0,1,0,0,0,1,0] => [[3,3,3,3],[2,1,1]] => [1,2,2,3] => [3,2,2,1]
[1,1,0,1,1,0,1,0,0,1,0,0] => [[4,3,3],[1,1]] => [3,2,3] => [3,2,3]
[1,1,0,1,1,0,1,0,1,0,0,0] => [[3,3,3,3],[1,1,1]] => [2,2,2,3] => [3,2,2,2]
[1,1,0,1,1,0,1,1,0,0,0,0] => [[4,4,3],[1,1]] => [3,3,3] => [3,3,3]
[1,1,0,1,1,1,0,0,0,0,1,0] => [[4,4,4],[3,1]] => [1,3,4] => [4,3,1]
[1,1,0,1,1,1,0,0,0,1,0,0] => [[5,4],[1]] => [4,4] => [4,4]
[1,1,0,1,1,1,0,0,1,0,0,0] => [[4,4,4],[2,1]] => [2,3,4] => [4,3,2]
[1,1,0,1,1,1,0,1,0,0,0,0] => [[5,5],[1]] => [4,5] => [5,4]
[1,1,0,1,1,1,1,0,0,0,0,0] => [[4,4,4],[1,1]] => [3,3,4] => [4,3,3]
[1,1,1,0,0,0,1,0,1,0,1,0] => [[2,2,2,2,2],[1,1,1]] => [1,1,1,2,2] => [2,2,1,1,1]
[1,1,1,0,0,0,1,0,1,1,0,0] => [[3,2,2,2],[1,1]] => [2,1,2,2] => [2,2,1,2]
[1,1,1,0,0,0,1,1,0,0,1,0] => [[3,3,2,2],[2,1]] => [1,2,2,2] => [2,2,2,1]
[1,1,1,0,0,0,1,1,0,1,0,0] => [[4,2,2],[1]] => [3,2,2] => [2,2,3]
[1,1,1,0,0,0,1,1,1,0,0,0] => [[3,3,2,2],[1,1]] => [2,2,2,2] => [2,2,2,2]
[1,1,1,0,0,1,0,0,1,0,1,0] => [[3,3,3,2],[2,2]] => [1,1,3,2] => [2,3,1,1]
[1,1,1,0,0,1,0,0,1,1,0,0] => [[4,3,2],[2]] => [2,3,2] => [2,3,2]
[1,1,1,0,0,1,0,1,0,0,1,0] => [[4,4,2],[3]] => [1,4,2] => [2,4,1]
[1,1,1,0,0,1,0,1,0,1,0,0] => [[5,2],[]] => [5,2] => [2,5]
[1,1,1,0,0,1,0,1,1,0,0,0] => [[4,4,2],[2]] => [2,4,2] => [2,4,2]
[1,1,1,0,0,1,1,0,0,0,1,0] => [[3,3,3,2],[2,1]] => [1,2,3,2] => [2,3,2,1]
[1,1,1,0,0,1,1,0,0,1,0,0] => [[4,3,2],[1]] => [3,3,2] => [2,3,3]
[1,1,1,0,0,1,1,0,1,0,0,0] => [[3,3,3,2],[1,1]] => [2,2,3,2] => [2,3,2,2]
[1,1,1,0,0,1,1,1,0,0,0,0] => [[4,4,2],[1]] => [3,4,2] => [2,4,3]
[1,1,1,0,1,0,0,0,1,0,1,0] => [[2,2,2,2,2],[1,1]] => [1,1,2,2,2] => [2,2,2,1,1]
[1,1,1,0,1,0,0,0,1,1,0,0] => [[3,2,2,2],[1]] => [2,2,2,2] => [2,2,2,2]
[1,1,1,0,1,0,0,1,0,0,1,0] => [[3,3,2,2],[2]] => [1,3,2,2] => [2,2,3,1]
[1,1,1,0,1,0,0,1,0,1,0,0] => [[4,2,2],[]] => [4,2,2] => [2,2,4]
[1,1,1,0,1,0,0,1,1,0,0,0] => [[3,3,2,2],[1]] => [2,3,2,2] => [2,2,3,2]
[1,1,1,0,1,0,1,0,0,0,1,0] => [[2,2,2,2,2],[1]] => [1,2,2,2,2] => [2,2,2,2,1]
[1,1,1,0,1,0,1,0,0,1,0,0] => [[3,2,2,2],[]] => [3,2,2,2] => [2,2,2,3]
[1,1,1,0,1,0,1,0,1,0,0,0] => [[2,2,2,2,2],[]] => [2,2,2,2,2] => [2,2,2,2,2]
[1,1,1,0,1,0,1,1,0,0,0,0] => [[3,3,2,2],[]] => [3,3,2,2] => [2,2,3,3]
[1,1,1,0,1,1,0,0,0,0,1,0] => [[3,3,3,2],[2]] => [1,3,3,2] => [2,3,3,1]
[1,1,1,0,1,1,0,0,0,1,0,0] => [[4,3,2],[]] => [4,3,2] => [2,3,4]
[1,1,1,0,1,1,0,0,1,0,0,0] => [[3,3,3,2],[1]] => [2,3,3,2] => [2,3,3,2]
[1,1,1,0,1,1,0,1,0,0,0,0] => [[4,4,2],[]] => [4,4,2] => [2,4,4]
[1,1,1,0,1,1,1,0,0,0,0,0] => [[3,3,3,2],[]] => [3,3,3,2] => [2,3,3,3]
[1,1,1,1,0,0,0,0,1,0,1,0] => [[3,3,3,3],[2,2]] => [1,1,3,3] => [3,3,1,1]
[1,1,1,1,0,0,0,0,1,1,0,0] => [[4,3,3],[2]] => [2,3,3] => [3,3,2]
[1,1,1,1,0,0,0,1,0,0,1,0] => [[4,4,3],[3]] => [1,4,3] => [3,4,1]
[1,1,1,1,0,0,0,1,0,1,0,0] => [[5,3],[]] => [5,3] => [3,5]
[1,1,1,1,0,0,0,1,1,0,0,0] => [[4,4,3],[2]] => [2,4,3] => [3,4,2]
[1,1,1,1,0,0,1,0,0,0,1,0] => [[3,3,3,3],[2,1]] => [1,2,3,3] => [3,3,2,1]
[1,1,1,1,0,0,1,0,0,1,0,0] => [[4,3,3],[1]] => [3,3,3] => [3,3,3]
[1,1,1,1,0,0,1,0,1,0,0,0] => [[3,3,3,3],[1,1]] => [2,2,3,3] => [3,3,2,2]
[1,1,1,1,0,0,1,1,0,0,0,0] => [[4,4,3],[1]] => [3,4,3] => [3,4,3]
[1,1,1,1,0,1,0,0,0,0,1,0] => [[4,4,4],[3]] => [1,4,4] => [4,4,1]
[1,1,1,1,0,1,0,0,0,1,0,0] => [[5,4],[]] => [5,4] => [4,5]
[1,1,1,1,0,1,0,0,1,0,0,0] => [[4,4,4],[2]] => [2,4,4] => [4,4,2]
[1,1,1,1,0,1,0,1,0,0,0,0] => [[5,5],[]] => [5,5] => [5,5]
[1,1,1,1,0,1,1,0,0,0,0,0] => [[4,4,4],[1]] => [3,4,4] => [4,4,3]
[1,1,1,1,1,0,0,0,0,0,1,0] => [[3,3,3,3],[2]] => [1,3,3,3] => [3,3,3,1]
[1,1,1,1,1,0,0,0,0,1,0,0] => [[4,3,3],[]] => [4,3,3] => [3,3,4]
[1,1,1,1,1,0,0,0,1,0,0,0] => [[3,3,3,3],[1]] => [2,3,3,3] => [3,3,3,2]
[1,1,1,1,1,0,0,1,0,0,0,0] => [[4,4,3],[]] => [4,4,3] => [3,4,4]
[1,1,1,1,1,0,1,0,0,0,0,0] => [[3,3,3,3],[]] => [3,3,3,3] => [3,3,3,3]
[1,1,1,1,1,1,0,0,0,0,0,0] => [[4,4,4],[]] => [4,4,4] => [4,4,4]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
[1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1,1],[]] => [2,1,1,1,1,1] => [1,1,1,1,1,2]
[1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[2,2,1,1,1,1],[1]] => [1,2,1,1,1,1] => [1,1,1,1,2,1]
[1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [[3,1,1,1,1],[]] => [3,1,1,1,1] => [1,1,1,1,3]
[1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1,1,1],[]] => [2,2,1,1,1,1] => [1,1,1,1,2,2]
[1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [[2,2,2,1,1,1],[1,1]] => [1,1,2,1,1,1] => [1,1,1,2,1,1]
[1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [[3,2,1,1,1],[1]] => [2,2,1,1,1] => [1,1,1,2,2]
[1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [[3,3,1,1,1],[2]] => [1,3,1,1,1] => [1,1,1,3,1]
[1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [[4,1,1,1],[]] => [4,1,1,1] => [1,1,1,4]
[1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [[3,2,1,1,1],[]] => [3,2,1,1,1] => [1,1,1,2,3]
[1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [[2,2,2,1,1,1],[]] => [2,2,2,1,1,1] => [1,1,1,2,2,2]
[1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[3,3,1,1,1],[]] => [3,3,1,1,1] => [1,1,1,3,3]
[1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1,2,1,1] => [1,1,2,1,1,1]
[1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [[3,2,2,1,1],[1,1]] => [2,1,2,1,1] => [1,1,2,1,2]
[1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[3,3,2,1,1],[2,1]] => [1,2,2,1,1] => [1,1,2,2,1]
[1,0,1,0,1,1,0,0,1,1,0,1,0,0] => [[4,2,1,1],[1]] => [3,2,1,1] => [1,1,2,3]
[1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [[3,3,3,1,1],[2,2]] => [1,1,3,1,1] => [1,1,3,1,1]
[1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [[4,3,1,1],[2]] => [2,3,1,1] => [1,1,3,2]
[1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [[4,4,1,1],[3]] => [1,4,1,1] => [1,1,4,1]
[1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1,1],[]] => [5,1,1] => [1,1,5]
[1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [[4,2,1,1],[]] => [4,2,1,1] => [1,1,2,4]
[1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [[3,2,2,1,1],[]] => [3,2,2,1,1] => [1,1,2,2,3]
[1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [[2,2,2,2,1,1],[]] => [2,2,2,2,1,1] => [1,1,2,2,2,2]
[1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [[3,3,2,1,1],[]] => [3,3,2,1,1] => [1,1,2,3,3]
[1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [[4,3,1,1],[]] => [4,3,1,1] => [1,1,3,4]
[1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [[4,4,1,1],[]] => [4,4,1,1] => [1,1,4,4]
[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1,2,1] => [1,2,1,1,1,1]
[1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [[3,2,2,2,1],[1,1,1]] => [2,1,1,2,1] => [1,2,1,1,2]
[1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [[3,3,2,2,1],[2,1,1]] => [1,2,1,2,1] => [1,2,1,2,1]
[1,0,1,1,0,0,1,0,1,1,0,1,0,0] => [[4,2,2,1],[1,1]] => [3,1,2,1] => [1,2,1,3]
[1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[3,3,3,2,1],[2,2,1]] => [1,1,2,2,1] => [1,2,2,1,1]
[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[4,3,2,1],[2,1]] => [2,2,2,1] => [1,2,2,2]
[1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [[4,4,2,1],[3,1]] => [1,3,2,1] => [1,2,3,1]
[1,0,1,1,0,0,1,1,0,1,0,1,0,0] => [[5,2,1],[1]] => [4,2,1] => [1,2,4]
[1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [[3,3,3,3,1],[2,2,2]] => [1,1,1,3,1] => [1,3,1,1,1]
[1,0,1,1,0,1,0,0,1,0,1,1,0,0] => [[4,3,3,1],[2,2]] => [2,1,3,1] => [1,3,1,2]
[1,0,1,1,0,1,0,0,1,1,0,0,1,0] => [[4,4,3,1],[3,2]] => [1,2,3,1] => [1,3,2,1]
[1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [[5,3,1],[2]] => [3,3,1] => [1,3,3]
[1,0,1,1,0,1,0,1,0,0,1,0,1,0] => [[4,4,4,1],[3,3]] => [1,1,4,1] => [1,4,1,1]
[1,0,1,1,0,1,0,1,0,0,1,1,0,0] => [[5,4,1],[3]] => [2,4,1] => [1,4,2]
[1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [[5,5,1],[4]] => [1,5,1] => [1,5,1]
[1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [[6,1],[]] => [6,1] => [1,6]
[1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [[5,2,1],[]] => [5,2,1] => [1,2,5]
[1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [[4,2,2,1],[]] => [4,2,2,1] => [1,2,2,4]
[1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [[3,2,2,2,1],[]] => [3,2,2,2,1] => [1,2,2,2,3]
[1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [[4,3,2,1],[]] => [4,3,2,1] => [1,2,3,4]
[1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [[3,3,3,2,1],[]] => [3,3,3,2,1] => [1,2,3,3,3]
[1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [[5,3,1],[]] => [5,3,1] => [1,3,5]
[1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [[5,4,1],[]] => [5,4,1] => [1,4,5]
[1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [[5,5,1],[]] => [5,5,1] => [1,5,5]
[1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [[4,4,4,1],[1]] => [3,4,4,1] => [1,4,4,3]
[1,0,1,1,1,1,1,0,0,0,1,0,0,0] => [[3,3,3,3,1],[1]] => [2,3,3,3,1] => [1,3,3,3,2]
[1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [[4,4,3,1],[]] => [4,4,3,1] => [1,3,4,4]
[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]] => [1,1,1,1,1,2] => [2,1,1,1,1,1]
[1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[3,2,2,2,2],[1,1,1,1]] => [2,1,1,1,2] => [2,1,1,1,2]
[1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[3,3,2,2,2],[2,1,1,1]] => [1,2,1,1,2] => [2,1,1,2,1]
[1,1,0,0,1,0,1,0,1,1,0,1,0,0] => [[4,2,2,2],[1,1,1]] => [3,1,1,2] => [2,1,1,3]
[1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[3,3,3,2,2],[2,2,1,1]] => [1,1,2,1,2] => [2,1,2,1,1]
[1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[4,3,2,2],[2,1,1]] => [2,2,1,2] => [2,1,2,2]
[1,1,0,0,1,0,1,1,0,1,0,0,1,0] => [[4,4,2,2],[3,1,1]] => [1,3,1,2] => [2,1,3,1]
[1,1,0,0,1,0,1,1,0,1,0,1,0,0] => [[5,2,2],[1,1]] => [4,1,2] => [2,1,4]
[1,1,0,0,1,1,0,0,1,0,1,0,1,0] => [[3,3,3,3,2],[2,2,2,1]] => [1,1,1,2,2] => [2,2,1,1,1]
[1,1,0,0,1,1,0,0,1,0,1,1,0,0] => [[4,3,3,2],[2,2,1]] => [2,1,2,2] => [2,2,1,2]
[1,1,0,0,1,1,0,0,1,1,0,0,1,0] => [[4,4,3,2],[3,2,1]] => [1,2,2,2] => [2,2,2,1]
[1,1,0,0,1,1,0,0,1,1,0,1,0,0] => [[5,3,2],[2,1]] => [3,2,2] => [2,2,3]
[1,1,0,0,1,1,0,1,0,0,1,0,1,0] => [[4,4,4,2],[3,3,1]] => [1,1,3,2] => [2,3,1,1]
[1,1,0,0,1,1,0,1,0,0,1,1,0,0] => [[5,4,2],[3,1]] => [2,3,2] => [2,3,2]
[1,1,0,0,1,1,0,1,0,1,0,0,1,0] => [[5,5,2],[4,1]] => [1,4,2] => [2,4,1]
[1,1,0,0,1,1,0,1,0,1,0,1,0,0] => [[6,2],[1]] => [5,2] => [2,5]
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [[4,4,4,2],[1,1,1]] => [3,3,3,2] => [2,3,3,3]
[1,1,0,1,0,0,1,0,1,0,1,0,1,0] => [[3,3,3,3,3],[2,2,2,2]] => [1,1,1,1,3] => [3,1,1,1,1]
[1,1,0,1,0,0,1,0,1,0,1,1,0,0] => [[4,3,3,3],[2,2,2]] => [2,1,1,3] => [3,1,1,2]
[1,1,0,1,0,0,1,0,1,1,0,0,1,0] => [[4,4,3,3],[3,2,2]] => [1,2,1,3] => [3,1,2,1]
[1,1,0,1,0,0,1,0,1,1,0,1,0,0] => [[5,3,3],[2,2]] => [3,1,3] => [3,1,3]
[1,1,0,1,0,0,1,1,0,0,1,0,1,0] => [[4,4,4,3],[3,3,2]] => [1,1,2,3] => [3,2,1,1]
[1,1,0,1,0,0,1,1,0,0,1,1,0,0] => [[5,4,3],[3,2]] => [2,2,3] => [3,2,2]
[1,1,0,1,0,0,1,1,0,1,0,0,1,0] => [[5,5,3],[4,2]] => [1,3,3] => [3,3,1]
[1,1,0,1,0,0,1,1,0,1,0,1,0,0] => [[6,3],[2]] => [4,3] => [3,4]
[1,1,0,1,0,1,0,0,1,0,1,0,1,0] => [[4,4,4,4],[3,3,3]] => [1,1,1,4] => [4,1,1,1]
[1,1,0,1,0,1,0,0,1,0,1,1,0,0] => [[5,4,4],[3,3]] => [2,1,4] => [4,1,2]
[1,1,0,1,0,1,0,0,1,1,0,0,1,0] => [[5,5,4],[4,3]] => [1,2,4] => [4,2,1]
[1,1,0,1,0,1,0,0,1,1,0,1,0,0] => [[6,4],[3]] => [3,4] => [4,3]
[1,1,0,1,0,1,0,1,0,0,1,0,1,0] => [[5,5,5],[4,4]] => [1,1,5] => [5,1,1]
[1,1,0,1,0,1,0,1,0,0,1,1,0,0] => [[6,5],[4]] => [2,5] => [5,2]
[1,1,0,1,0,1,0,1,0,1,0,0,1,0] => [[6,6],[5]] => [1,6] => [6,1]
[1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[7],[]] => [7] => [7]
[1,1,0,1,0,1,0,1,0,1,1,0,0,0] => [[6,6],[4]] => [2,6] => [6,2]
[1,1,0,1,0,1,0,1,1,0,0,1,0,0] => [[6,5],[3]] => [3,5] => [5,3]
[1,1,0,1,0,1,0,1,1,0,1,0,0,0] => [[5,5,5],[3,3]] => [2,2,5] => [5,2,2]
[1,1,0,1,0,1,0,1,1,1,0,0,0,0] => [[6,6],[3]] => [3,6] => [6,3]
[1,1,0,1,0,1,1,0,0,1,0,1,0,0] => [[6,4],[2]] => [4,4] => [4,4]
[1,1,0,1,0,1,1,0,0,1,1,0,0,0] => [[5,5,4],[3,2]] => [2,3,4] => [4,3,2]
[1,1,0,1,0,1,1,0,1,0,0,1,0,0] => [[5,4,4],[2,2]] => [3,2,4] => [4,2,3]
[1,1,0,1,0,1,1,0,1,0,1,0,0,0] => [[4,4,4,4],[2,2,2]] => [2,2,2,4] => [4,2,2,2]
[1,1,0,1,0,1,1,0,1,1,0,0,0,0] => [[5,5,4],[2,2]] => [3,3,4] => [4,3,3]
[1,1,0,1,0,1,1,1,0,0,0,1,0,0] => [[6,5],[2]] => [4,5] => [5,4]
[1,1,0,1,0,1,1,1,0,0,1,0,0,0] => [[5,5,5],[3,2]] => [2,3,5] => [5,3,2]
[1,1,0,1,0,1,1,1,0,1,0,0,0,0] => [[6,6],[2]] => [4,6] => [6,4]
[1,1,0,1,1,0,0,1,0,1,0,1,0,0] => [[6,3],[1]] => [5,3] => [3,5]
[1,1,0,1,1,0,0,1,0,1,1,0,0,0] => [[5,5,3],[3,1]] => [2,4,3] => [3,4,2]
[1,1,0,1,1,0,0,1,1,0,0,1,0,0] => [[5,4,3],[2,1]] => [3,3,3] => [3,3,3]
[1,1,0,1,1,0,0,1,1,0,1,0,0,0] => [[4,4,4,3],[2,2,1]] => [2,2,3,3] => [3,3,2,2]
[1,1,0,1,1,0,0,1,1,1,0,0,0,0] => [[5,5,3],[2,1]] => [3,4,3] => [3,4,3]
[1,1,0,1,1,0,1,0,0,1,0,1,0,0] => [[5,3,3],[1,1]] => [4,2,3] => [3,2,4]
[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [[4,4,3,3],[2,1,1]] => [2,3,2,3] => [3,2,3,2]
[1,1,0,1,1,0,1,0,1,0,0,1,0,0] => [[4,3,3,3],[1,1,1]] => [3,2,2,3] => [3,2,2,3]
[1,1,0,1,1,0,1,1,0,0,0,1,0,0] => [[5,4,3],[1,1]] => [4,3,3] => [3,3,4]
[1,1,0,1,1,0,1,1,0,0,1,0,0,0] => [[4,4,4,3],[2,1,1]] => [2,3,3,3] => [3,3,3,2]
[1,1,0,1,1,0,1,1,0,1,0,0,0,0] => [[5,5,3],[1,1]] => [4,4,3] => [3,4,4]
[1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [[4,4,4,3],[1,1,1]] => [3,3,3,3] => [3,3,3,3]
[1,1,0,1,1,1,0,0,0,1,0,1,0,0] => [[6,4],[1]] => [5,4] => [4,5]
[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [[5,5,4],[3,1]] => [2,4,4] => [4,4,2]
[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [[5,4,4],[2,1]] => [3,3,4] => [4,3,3]
[1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [[5,5,4],[2,1]] => [3,4,4] => [4,4,3]
[1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [[6,5],[1]] => [5,5] => [5,5]
[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [[5,5,5],[2,1]] => [3,4,5] => [5,4,3]
[1,1,0,1,1,1,1,0,0,0,1,0,0,0] => [[4,4,4,4],[2,1,1]] => [2,3,3,4] => [4,3,3,2]
[1,1,0,1,1,1,1,0,0,1,0,0,0,0] => [[5,5,4],[1,1]] => [4,4,4] => [4,4,4]
[1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [[6,2],[]] => [6,2] => [2,6]
[1,1,1,0,0,1,0,1,0,1,1,0,0,0] => [[5,5,2],[3]] => [2,5,2] => [2,5,2]
[1,1,1,0,0,1,0,1,1,0,0,1,0,0] => [[5,4,2],[2]] => [3,4,2] => [2,4,3]
[1,1,1,0,0,1,0,1,1,0,1,0,0,0] => [[4,4,4,2],[2,2]] => [2,2,4,2] => [2,4,2,2]
[1,1,1,0,0,1,0,1,1,1,0,0,0,0] => [[5,5,2],[2]] => [3,5,2] => [2,5,3]
[1,1,1,0,0,1,1,0,0,1,0,1,0,0] => [[5,3,2],[1]] => [4,3,2] => [2,3,4]
[1,1,1,0,0,1,1,0,0,1,1,0,0,0] => [[4,4,3,2],[2,1]] => [2,3,3,2] => [2,3,3,2]
[1,1,1,0,0,1,1,0,1,0,0,1,0,0] => [[4,3,3,2],[1,1]] => [3,2,3,2] => [2,3,2,3]
[1,1,1,0,0,1,1,0,1,1,0,0,0,0] => [[4,4,3,2],[1,1]] => [3,3,3,2] => [2,3,3,3]
[1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [[5,4,2],[1]] => [4,4,2] => [2,4,4]
[1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [[4,4,4,2],[1,1]] => [3,3,4,2] => [2,4,3,3]
[1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [[5,2,2],[]] => [5,2,2] => [2,2,5]
[1,1,1,0,1,0,0,1,0,1,1,0,0,0] => [[4,4,2,2],[2]] => [2,4,2,2] => [2,2,4,2]
[1,1,1,0,1,0,0,1,1,0,0,1,0,0] => [[4,3,2,2],[1]] => [3,3,2,2] => [2,2,3,3]
[1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [[4,2,2,2],[]] => [4,2,2,2] => [2,2,2,4]
[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [[2,2,2,2,2,2],[]] => [2,2,2,2,2,2] => [2,2,2,2,2,2]
[1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [[3,3,2,2,2],[]] => [3,3,2,2,2] => [2,2,2,3,3]
[1,1,1,0,1,0,1,1,0,0,1,0,0,0] => [[3,3,3,2,2],[1]] => [2,3,3,2,2] => [2,2,3,3,2]
[1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [[4,4,2,2],[]] => [4,4,2,2] => [2,2,4,4]
[1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [[5,3,2],[]] => [5,3,2] => [2,3,5]
[1,1,1,0,1,1,0,0,1,0,0,1,0,0] => [[4,3,3,2],[1]] => [3,3,3,2] => [2,3,3,3]
[1,1,1,0,1,1,0,0,1,0,1,0,0,0] => [[3,3,3,3,2],[1,1]] => [2,2,3,3,2] => [2,3,3,2,2]
[1,1,1,0,1,1,0,0,1,1,0,0,0,0] => [[4,4,3,2],[1]] => [3,4,3,2] => [2,3,4,3]
[1,1,1,0,1,1,0,1,0,0,1,0,0,0] => [[4,4,4,2],[2]] => [2,4,4,2] => [2,4,4,2]
[1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [[5,5,2],[]] => [5,5,2] => [2,5,5]
[1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [[3,3,3,3,2],[2]] => [1,3,3,3,2] => [2,3,3,3,1]
[1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [[4,3,3,2],[]] => [4,3,3,2] => [2,3,3,4]
[1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [[6,3],[]] => [6,3] => [3,6]
[1,1,1,1,0,0,0,1,0,1,1,0,0,0] => [[5,5,3],[3]] => [2,5,3] => [3,5,2]
[1,1,1,1,0,0,0,1,1,0,0,1,0,0] => [[5,4,3],[2]] => [3,4,3] => [3,4,3]
[1,1,1,1,0,0,1,0,0,1,0,1,0,0] => [[5,3,3],[1]] => [4,3,3] => [3,3,4]
[1,1,1,1,0,0,1,0,0,1,1,0,0,0] => [[4,4,3,3],[2,1]] => [2,3,3,3] => [3,3,3,2]
[1,1,1,1,0,0,1,0,1,0,1,0,0,0] => [[3,3,3,3,3],[1,1,1]] => [2,2,2,3,3] => [3,3,2,2,2]
[1,1,1,1,0,0,1,0,1,1,0,0,0,0] => [[4,4,3,3],[1,1]] => [3,3,3,3] => [3,3,3,3]
[1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [[5,4,3],[1]] => [4,4,3] => [3,4,4]
[1,1,1,1,0,0,1,1,0,0,1,0,0,0] => [[4,4,4,3],[2,1]] => [2,3,4,3] => [3,4,3,2]
[1,1,1,1,0,0,1,1,0,1,0,0,0,0] => [[5,5,3],[1]] => [4,5,3] => [3,5,4]
[1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [[6,4],[]] => [6,4] => [4,6]
[1,1,1,1,0,1,0,0,1,0,0,1,0,0] => [[5,4,4],[2]] => [3,4,4] => [4,4,3]
[1,1,1,1,0,1,0,0,1,0,1,0,0,0] => [[4,4,4,4],[2,2]] => [2,2,4,4] => [4,4,2,2]
[1,1,1,1,0,1,0,0,1,1,0,0,0,0] => [[5,5,4],[2]] => [3,5,4] => [4,5,3]
[1,1,1,1,0,1,0,1,0,0,0,0,1,0] => [[5,5,5],[4]] => [1,5,5] => [5,5,1]
[1,1,1,1,0,1,0,1,0,0,1,0,0,0] => [[5,5,5],[3]] => [2,5,5] => [5,5,2]
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [[6,6],[]] => [6,6] => [6,6]
[1,1,1,1,0,1,1,0,0,0,0,0,1,0] => [[4,4,4,4],[3,1]] => [1,3,4,4] => [4,4,3,1]
[1,1,1,1,0,1,1,0,0,0,0,1,0,0] => [[5,4,4],[1]] => [4,4,4] => [4,4,4]
[1,1,1,1,1,0,0,0,0,0,1,1,0,0] => [[4,3,3,3],[2]] => [2,3,3,3] => [3,3,3,2]
[1,1,1,1,1,0,0,0,0,1,1,0,0,0] => [[4,4,3,3],[2]] => [2,4,3,3] => [3,3,4,2]
[1,1,1,1,1,0,0,0,1,0,0,0,1,0] => [[3,3,3,3,3],[2,1]] => [1,2,3,3,3] => [3,3,3,2,1]
[1,1,1,1,1,0,0,0,1,0,0,1,0,0] => [[4,3,3,3],[1]] => [3,3,3,3] => [3,3,3,3]
[1,1,1,1,1,0,0,1,0,0,0,0,1,0] => [[4,4,4,3],[3]] => [1,4,4,3] => [3,4,4,1]
[1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [[5,4,3],[]] => [5,4,3] => [3,4,5]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1,1,1],[]] => [2,1,1,1,1,1,1] => [1,1,1,1,1,1,2]
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [[3,1,1,1,1,1],[]] => [3,1,1,1,1,1] => [1,1,1,1,1,3]
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1,1,1,1],[]] => [2,2,1,1,1,1,1] => [1,1,1,1,1,2,2]
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [[4,1,1,1,1],[]] => [4,1,1,1,1] => [1,1,1,1,4]
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [[3,2,1,1,1,1],[]] => [3,2,1,1,1,1] => [1,1,1,1,2,3]
[1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [[2,2,2,1,1,1,1],[]] => [2,2,2,1,1,1,1] => [1,1,1,1,2,2,2]
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[3,3,1,1,1,1],[]] => [3,3,1,1,1,1] => [1,1,1,1,3,3]
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1,1,1],[]] => [5,1,1,1] => [1,1,1,5]
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [[4,2,1,1,1],[]] => [4,2,1,1,1] => [1,1,1,2,4]
[1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [[3,2,2,1,1,1],[]] => [3,2,2,1,1,1] => [1,1,1,2,2,3]
[1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [[4,3,1,1,1],[]] => [4,3,1,1,1] => [1,1,1,3,4]
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [[6,1,1],[]] => [6,1,1] => [1,1,6]
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [[5,2,1,1],[]] => [5,2,1,1] => [1,1,2,5]
[1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [[4,2,2,1,1],[]] => [4,2,2,1,1] => [1,1,2,2,4]
[1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [[5,3,1,1],[]] => [5,3,1,1] => [1,1,3,5]
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[7,1],[]] => [7,1] => [1,7]
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [[6,2,1],[]] => [6,2,1] => [1,2,6]
[1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [[5,2,2,1],[]] => [5,2,2,1] => [1,2,2,5]
[1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [[6,3,1],[]] => [6,3,1] => [1,3,6]
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[8],[]] => [8] => [8]
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [[7,2],[]] => [7,2] => [2,7]
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0] => [[6,2,2],[]] => [6,2,2] => [2,2,6]
[1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0] => [[7,3],[]] => [7,3] => [3,7]
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[9],[]] => [9] => [9]
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[8,2],[]] => [8,2] => [2,8]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1,1,1,1],[]] => [2,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,2]
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [[3,1,1,1,1,1,1],[]] => [3,1,1,1,1,1,1] => [1,1,1,1,1,1,3]
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [[4,1,1,1,1,1],[]] => [4,1,1,1,1,1] => [1,1,1,1,1,4]
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1,1,1,1],[]] => [5,1,1,1,1] => [1,1,1,1,5]
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [[6,1,1,1],[]] => [6,1,1,1] => [1,1,1,6]
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[7,1,1],[]] => [7,1,1] => [1,1,7]
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[8,1],[]] => [8,1] => [1,8]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,1,1]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1,1,1,1,1],[]] => [2,1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1,2]
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1,1,1,1,1],[]] => [2,2,1,1,1,1,1,1] => [1,1,1,1,1,1,2,2]
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [[3,1,1,1,1,1,1,1],[]] => [3,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,3]
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [[3,2,1,1,1,1,1],[]] => [3,2,1,1,1,1,1] => [1,1,1,1,1,2,3]
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [[4,1,1,1,1,1,1],[]] => [4,1,1,1,1,1,1] => [1,1,1,1,1,1,4]
[1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [[4,2,1,1,1,1],[]] => [4,2,1,1,1,1] => [1,1,1,1,2,4]
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1,1,1,1,1],[]] => [5,1,1,1,1,1] => [1,1,1,1,1,5]
[1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [[5,2,1,1,1],[]] => [5,2,1,1,1] => [1,1,1,2,5]
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [[6,1,1,1,1],[]] => [6,1,1,1,1] => [1,1,1,1,6]
[1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [[6,2,1,1],[]] => [6,2,1,1] => [1,1,2,6]
[1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[7,1,1,1],[]] => [7,1,1,1] => [1,1,1,7]
[1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [[7,2,1],[]] => [7,2,1] => [1,2,7]
[1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[8,1,1],[]] => [8,1,1] => [1,1,8]
[1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[9,1],[]] => [9,1] => [1,9]
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[10],[]] => [10] => [10]
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
row lengths
Description
The sequence of row lengths from top to bottom.
Map
reverse
Description
Return the reversal of a composition.
That is, the composition $(i_1, i_2, \ldots, i_k)$ is sent to $(i_k, i_{k-1}, \ldots, i_1)$.