Identifier
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Images
[1] => [1,0] => [[1],[]]
[2] => [1,0,1,0] => [[1,1],[]]
[1,1] => [1,1,0,0] => [[2],[]]
[3] => [1,0,1,0,1,0] => [[1,1,1],[]]
[2,1] => [1,0,1,1,0,0] => [[2,1],[]]
[1,1,1] => [1,1,0,1,0,0] => [[3],[]]
[4] => [1,0,1,0,1,0,1,0] => [[1,1,1,1],[]]
[3,1] => [1,0,1,0,1,1,0,0] => [[2,1,1],[]]
[2,2] => [1,1,1,0,0,0] => [[2,2],[]]
[2,1,1] => [1,0,1,1,0,1,0,0] => [[3,1],[]]
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [[4],[]]
[5] => [1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1],[]]
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1],[]]
[3,2] => [1,0,1,1,1,0,0,0] => [[2,2,1],[]]
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [[3,1,1],[]]
[2,2,1] => [1,1,1,0,0,1,0,0] => [[3,2],[]]
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [[4,1],[]]
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [[5],[]]
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1,1],[]]
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1],[]]
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1],[]]
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [[3,1,1,1],[]]
[3,3] => [1,1,1,0,1,0,0,0] => [[2,2,2],[]]
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [[3,2,1],[]]
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [[4,1,1],[]]
[2,2,2] => [1,1,1,1,0,0,0,0] => [[3,3],[]]
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [[4,2],[]]
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1],[]]
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [[6],[]]
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1,1,1],[]]
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1,1],[]]
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1,1],[]]
[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [[3,1,1,1,1],[]]
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [[2,2,2,1],[]]
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [[3,2,1,1],[]]
[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [[4,1,1,1],[]]
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [[3,2,2],[]]
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [[3,3,1],[]]
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [[4,2,1],[]]
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1,1],[]]
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [[4,3],[]]
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [[5,2],[]]
[2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [[6,1],[]]
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[7],[]]
[8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[1,1,1,1,1,1,1,1],[]]
[7,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1,1,1],[]]
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1,1,1],[]]
[6,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [[3,1,1,1,1,1],[]]
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [[2,2,2,1,1],[]]
[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [[3,2,1,1,1],[]]
[5,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [[4,1,1,1,1],[]]
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [[2,2,2,2],[]]
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [[3,2,2,1],[]]
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [[3,3,1,1],[]]
[4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [[4,2,1,1],[]]
[4,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1,1,1],[]]
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [[3,3,2],[]]
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [[4,2,2],[]]
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [[4,3,1],[]]
[3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [[5,2,1],[]]
[3,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [[6,1,1],[]]
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [[4,4],[]]
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [[5,3],[]]
[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [[6,2],[]]
[2,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[7,1],[]]
[1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[8],[]]
[9] => [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],[]]
[8,1] => [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],[]]
[7,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1,1,1,1],[]]
[7,1,1] => [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],[]]
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [[2,2,2,1,1,1],[]]
[6,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [[3,2,1,1,1,1],[]]
[6,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [[4,1,1,1,1,1],[]]
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [[2,2,2,2,1],[]]
[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [[3,2,2,1,1],[]]
[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[3,3,1,1,1],[]]
[5,2,1,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [[4,2,1,1,1],[]]
[5,1,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [[5,1,1,1,1],[]]
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [[3,2,2,2],[]]
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [[3,3,2,1],[]]
[4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [[4,2,2,1],[]]
[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [[4,3,1,1],[]]
[4,2,1,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [[5,2,1,1],[]]
[4,1,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [[6,1,1,1],[]]
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [[3,3,3],[]]
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [[4,3,2],[]]
[3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [[5,2,2],[]]
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [[4,4,1],[]]
[3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [[5,3,1],[]]
[3,2,1,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [[6,2,1],[]]
[3,1,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[7,1,1],[]]
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [[5,4],[]]
[2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [[6,3],[]]
[2,2,1,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [[7,2],[]]
[2,1,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[8,1],[]]
[1,1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[9],[]]
[10] => [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],[]]
[9,1] => [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],[]]
[8,2] => [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],[]]
[8,1,1] => [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],[]]
[7,3] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [[2,2,2,1,1,1,1],[]]
>>> Load all 337 entries. <<<
[7,2,1] => [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],[]]
[7,1,1,1] => [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],[]]
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [[2,2,2,2,1,1],[]]
[6,3,1] => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [[3,2,2,1,1,1],[]]
[6,2,2] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[3,3,1,1,1,1],[]]
[6,2,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [[4,2,1,1,1,1],[]]
[6,1,1,1,1] => [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],[]]
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [[2,2,2,2,2],[]]
[5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [[3,2,2,2,1],[]]
[5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [[3,3,2,1,1],[]]
[5,3,1,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [[4,2,2,1,1],[]]
[5,2,2,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [[4,3,1,1,1],[]]
[5,2,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [[5,2,1,1,1],[]]
[5,1,1,1,1,1] => [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],[]]
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [[3,3,2,2],[]]
[4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [[4,2,2,2],[]]
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [[3,3,3,1],[]]
[4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [[4,3,2,1],[]]
[4,3,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [[5,2,2,1],[]]
[4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [[4,4,1,1],[]]
[4,2,2,1,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [[5,3,1,1],[]]
[4,2,1,1,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [[6,2,1,1],[]]
[4,1,1,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[7,1,1,1],[]]
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [[4,3,3],[]]
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [[4,4,2],[]]
[3,3,2,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [[5,3,2],[]]
[3,3,1,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0] => [[6,2,2],[]]
[3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [[5,4,1],[]]
[3,2,2,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [[6,3,1],[]]
[3,2,1,1,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [[7,2,1],[]]
[3,1,1,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[8,1,1],[]]
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [[5,5],[]]
[2,2,2,2,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [[6,4],[]]
[2,2,2,1,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0] => [[7,3],[]]
[2,2,1,1,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[8,2],[]]
[2,1,1,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[9,1],[]]
[1,1,1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[10],[]]
[10,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[2,1,1,1,1,1,1,1,1,1],[]]
[9,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[2,2,1,1,1,1,1,1,1],[]]
[9,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [[3,1,1,1,1,1,1,1,1],[]]
[8,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [[2,2,2,1,1,1,1,1],[]]
[7,4] => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [[2,2,2,2,1,1,1],[]]
[7,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[3,3,1,1,1,1,1],[]]
[6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [[2,2,2,2,2,1],[]]
[6,4,1] => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [[3,2,2,2,1,1],[]]
[6,3,2] => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [[3,3,2,1,1,1],[]]
[5,5,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [[3,2,2,2,2],[]]
[5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [[3,3,2,2,1],[]]
[5,4,1,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [[4,2,2,2,1],[]]
[5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [[3,3,3,1,1],[]]
[5,3,2,1] => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [[4,3,2,1,1],[]]
[5,2,2,2] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [[4,4,1,1,1],[]]
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [[3,3,3,2],[]]
[4,4,2,1] => [1,1,1,0,1,0,1,1,0,0,0,1,0,0] => [[4,3,2,2],[]]
[4,4,1,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0] => [[5,2,2,2],[]]
[4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [[4,3,3,1],[]]
[4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [[4,4,2,1],[]]
[4,3,2,1,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [[5,3,2,1],[]]
[4,2,2,2,1] => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [[5,4,1,1],[]]
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [[4,4,3],[]]
[3,3,3,1,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [[5,3,3],[]]
[3,3,2,2,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0] => [[5,4,2],[]]
[3,3,2,1,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0] => [[6,3,2],[]]
[3,3,1,1,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [[7,2,2],[]]
[3,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [[5,5,1],[]]
[3,2,2,2,1,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [[6,4,1],[]]
[2,2,2,2,2,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [[6,5],[]]
[2,2,2,2,1,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0] => [[7,4],[]]
[2,2,2,1,1,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0] => [[8,3],[]]
[2,2,1,1,1,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[9,2],[]]
[1,1,1,1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[11],[]]
[8,4] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [[2,2,2,2,1,1,1,1],[]]
[8,2,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[3,3,1,1,1,1,1,1],[]]
[7,5] => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [[2,2,2,2,2,1,1],[]]
[6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [[2,2,2,2,2,2],[]]
[6,5,1] => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [[3,2,2,2,2,1],[]]
[6,4,2] => [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [[3,3,2,2,1,1],[]]
[6,4,1,1] => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [[4,2,2,2,1,1],[]]
[6,3,3] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [[3,3,3,1,1,1],[]]
[5,5,2] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [[3,3,2,2,2],[]]
[5,5,1,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0] => [[4,2,2,2,2],[]]
[5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [[3,3,3,2,1],[]]
[5,4,2,1] => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0] => [[4,3,2,2,1],[]]
[5,3,3,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [[4,3,3,1,1],[]]
[5,3,2,2] => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [[4,4,2,1,1],[]]
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [[3,3,3,3],[]]
[4,4,3,1] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [[4,3,3,2],[]]
[4,4,2,2] => [1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [[4,4,2,2],[]]
[4,4,2,1,1] => [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0] => [[5,3,2,2],[]]
[4,4,1,1,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0] => [[6,2,2,2],[]]
[4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [[4,4,3,1],[]]
[4,3,3,1,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [[5,3,3,1],[]]
[4,3,2,2,1] => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0] => [[5,4,2,1],[]]
[4,2,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [[5,5,1,1],[]]
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [[4,4,4],[]]
[3,3,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [[5,4,3],[]]
[3,3,3,1,1,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0] => [[6,3,3],[]]
[3,3,2,2,2] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [[5,5,2],[]]
[3,3,2,2,1,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0] => [[6,4,2],[]]
[3,2,2,2,2,1] => [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [[6,5,1],[]]
[2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [[6,6],[]]
[2,2,2,2,2,1,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0] => [[7,5],[]]
[2,2,2,2,1,1,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0] => [[8,4],[]]
[2,2,2,1,1,1,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [[9,3],[]]
[8,5] => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [[2,2,2,2,2,1,1,1],[]]
[7,6] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [[2,2,2,2,2,2,1],[]]
[7,3,3] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [[3,3,3,1,1,1,1],[]]
[6,6,1] => [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0] => [[3,2,2,2,2,2],[]]
[6,5,2] => [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [[3,3,2,2,2,1],[]]
[6,4,3] => [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [[3,3,3,2,1,1],[]]
[5,5,3] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [[3,3,3,2,2],[]]
[5,5,2,1] => [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0] => [[4,3,2,2,2],[]]
[5,4,4] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [[3,3,3,3,1],[]]
[5,4,3,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [[4,3,3,2,1],[]]
[5,4,2,2] => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [[4,4,2,2,1],[]]
[5,3,3,2] => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [[4,4,3,1,1],[]]
[4,4,4,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [[4,3,3,3],[]]
[4,4,3,2] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [[4,4,3,2],[]]
[4,4,3,1,1] => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0] => [[5,3,3,2],[]]
[4,4,2,2,1] => [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0] => [[5,4,2,2],[]]
[4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[4,4,4,1],[]]
[4,3,3,2,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [[5,4,3,1],[]]
[4,3,2,2,2] => [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [[5,5,2,1],[]]
[3,3,3,3,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [[5,4,4],[]]
[3,3,3,2,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [[5,5,3],[]]
[3,3,3,2,1,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0] => [[6,4,3],[]]
[3,3,2,2,2,1] => [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0] => [[6,5,2],[]]
[3,2,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [[6,6,1],[]]
[2,2,2,2,2,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0] => [[7,6],[]]
[2,2,2,2,2,1,1,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,0] => [[8,5],[]]
[2,2,2,2,1,1,1,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0] => [[9,4],[]]
[8,6] => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [[2,2,2,2,2,2,1,1],[]]
[8,3,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [[3,3,3,1,1,1,1,1],[]]
[7,7] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [[2,2,2,2,2,2,2],[]]
[7,6,1] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0] => [[3,2,2,2,2,2,1],[]]
[6,6,2] => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0] => [[3,3,2,2,2,2],[]]
[6,5,3] => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [[3,3,3,2,2,1],[]]
[6,4,4] => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [[3,3,3,3,1,1],[]]
[5,5,4] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [[3,3,3,3,2],[]]
[5,5,3,1] => [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0] => [[4,3,3,2,2],[]]
[5,5,2,2] => [1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0] => [[4,4,2,2,2],[]]
[5,5,1,1,1,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0] => [[6,2,2,2,2],[]]
[5,4,4,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [[4,3,3,3,1],[]]
[5,4,3,2] => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [[4,4,3,2,1],[]]
[5,3,3,3] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[4,4,4,1,1],[]]
[4,4,4,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [[4,4,3,3],[]]
[4,4,4,1,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0] => [[5,3,3,3],[]]
[4,4,3,3] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [[4,4,4,2],[]]
[4,4,3,2,1] => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0] => [[5,4,3,2],[]]
[4,4,2,2,2] => [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0] => [[5,5,2,2],[]]
[4,3,3,3,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [[5,4,4,1],[]]
[4,3,3,2,2] => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [[5,5,3,1],[]]
[3,3,3,3,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [[5,5,4],[]]
[3,3,3,3,1,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0] => [[6,4,4],[]]
[3,3,3,2,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0] => [[6,5,3],[]]
[3,3,2,2,2,2] => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0] => [[6,6,2],[]]
[2,2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0] => [[7,7],[]]
[2,2,2,2,2,2,1,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,0] => [[8,6],[]]
[2,2,2,2,2,1,1,1,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0] => [[9,5],[]]
[8,7] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [[2,2,2,2,2,2,2,1],[]]
[7,7,1] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0] => [[3,2,2,2,2,2,2],[]]
[6,6,3] => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0] => [[3,3,3,2,2,2],[]]
[6,5,4] => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [[3,3,3,3,2,1],[]]
[6,3,3,3] => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[4,4,4,1,1,1],[]]
[5,5,5] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [[3,3,3,3,3],[]]
[5,5,4,1] => [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0] => [[4,3,3,3,2],[]]
[5,5,3,2] => [1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,0] => [[4,4,3,2,2],[]]
[5,4,4,2] => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [[4,4,3,3,1],[]]
[5,4,3,3] => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [[4,4,4,2,1],[]]
[4,4,4,3] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [[4,4,4,3],[]]
[4,4,4,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0] => [[5,4,3,3],[]]
[4,4,3,3,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0] => [[5,4,4,2],[]]
[4,4,3,2,2] => [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0] => [[5,5,3,2],[]]
[4,3,3,3,2] => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [[5,5,4,1],[]]
[3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [[5,5,5],[]]
[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],[]]
[3,3,3,2,2,2] => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0] => [[6,6,3],[]]
[3,2,2,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0] => [[7,7,1],[]]
[2,2,2,2,2,2,2,1] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0] => [[8,7],[]]
[2,2,2,2,2,2,1,1,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0,0] => [[9,6],[]]
[8,8] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [[2,2,2,2,2,2,2,2],[]]
[7,3,3,3] => [1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[4,4,4,1,1,1,1],[]]
[6,6,4] => [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0] => [[3,3,3,3,2,2],[]]
[6,5,5] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [[3,3,3,3,3,1],[]]
[5,5,5,1] => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0] => [[4,3,3,3,3],[]]
[5,5,4,2] => [1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,0] => [[4,4,3,3,2],[]]
[5,5,3,3] => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0] => [[4,4,4,2,2],[]]
[5,5,2,2,2] => [1,1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0,0] => [[5,5,2,2,2],[]]
[5,4,4,3] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [[4,4,4,3,1],[]]
[4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [[4,4,4,4],[]]
[4,4,4,3,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0] => [[5,4,4,3],[]]
[4,4,4,2,2] => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0] => [[5,5,3,3],[]]
[4,4,3,3,2] => [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0] => [[5,5,4,2],[]]
[4,3,3,3,3] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [[5,5,5,1],[]]
[4,3,3,2,2,2] => [1,0,1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0] => [[6,6,3,1],[]]
[3,3,3,3,3,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0] => [[6,5,5],[]]
[3,3,3,3,2,2] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0] => [[6,6,4],[]]
[2,2,2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0] => [[8,8],[]]
[2,2,2,2,2,2,2,1,1] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0,0] => [[9,7],[]]
[9,8] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [[2,2,2,2,2,2,2,2,1],[]]
[8,8,1] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0] => [[3,2,2,2,2,2,2,2],[]]
[6,6,5] => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0] => [[3,3,3,3,3,2],[]]
[5,5,5,2] => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0] => [[4,4,3,3,3],[]]
[5,5,4,3] => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0] => [[4,4,4,3,2],[]]
[5,4,4,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [[4,4,4,4,1],[]]
[4,4,4,4,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [[5,4,4,4],[]]
[4,4,4,3,2] => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [[5,5,4,3],[]]
[4,4,3,3,3] => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0] => [[5,5,5,2],[]]
[3,3,3,3,3,2] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [[6,6,5],[]]
[3,3,3,2,2,2,2] => [1,1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0,0] => [[7,7,3],[]]
[3,2,2,2,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0] => [[8,8,1],[]]
[2,2,2,2,2,2,2,2,1] => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0,0] => [[9,8],[]]
[6,5,4,2,1] => [1,0,1,1,1,0,1,1,1,0,1,0,0,1,0,0,0,1,0,0] => [[5,4,3,3,2,1],[]]
[6,5,4,3,3,2,1] => [1,0,1,1,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0,0,1,0,0] => [[7,6,5,3,2,1],[]]
[4,4,4,4,3,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0] => [[6,5,5,4],[]]
[5,4,4,4,3] => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [[5,5,5,4,1],[]]
[4,4,4,4,2] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [[5,5,4,4],[]]
[2,2,2,2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0] => [[9,9],[]]
[3,3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [[6,6,6],[]]
[3,3,3,3,3,3,1] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0,0] => [[7,6,6],[]]
[4,3,3,3,3,3] => [1,0,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [[6,6,6,1],[]]
[4,4,4,4,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [[5,5,5,5],[]]
[4,4,4,4,4,4] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [[6,6,6,6],[]]
[5,5,5,5] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [[4,4,4,4,4],[]]
[6,6,6,6] => [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [[4,4,4,4,4,4],[]]
[6,6,6] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [[3,3,3,3,3,3],[]]
[7,6,6] => [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [[3,3,3,3,3,3,1],[]]
[10,10] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [[2,2,2,2,2,2,2,2,2,2],[]]
[6,6,6,6,6] => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [[5,5,5,5,5,5],[]]
[5,5,5,5,5,5] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [[6,6,6,6,6],[]]
[5,5,5,5,5] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [[5,5,5,5,5],[]]
[9,9] => [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [[2,2,2,2,2,2,2,2,2],[]]
[6,5,5,5] => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [[4,4,4,4,4,1],[]]
[5,4,4,4,4] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [[5,5,5,5,1],[]]
[4,4,4,4,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0] => [[6,5,5,5],[]]
[5,5,5,5,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0] => [[5,4,4,4,4],[]]
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
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 $\gamma(D)$.