Identifier
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00187: Skew partitions —conjugate⟶ Skew partitions
Images
[1] => [[1],[]] => [[1],[]]
[2] => [[2],[]] => [[1,1],[]]
[1,1] => [[1,1],[]] => [[2],[]]
[3] => [[3],[]] => [[1,1,1],[]]
[2,1] => [[2,1],[]] => [[2,1],[]]
[1,1,1] => [[1,1,1],[]] => [[3],[]]
[4] => [[4],[]] => [[1,1,1,1],[]]
[3,1] => [[3,1],[]] => [[2,1,1],[]]
[2,2] => [[2,2],[]] => [[2,2],[]]
[2,1,1] => [[2,1,1],[]] => [[3,1],[]]
[1,1,1,1] => [[1,1,1,1],[]] => [[4],[]]
[5] => [[5],[]] => [[1,1,1,1,1],[]]
[4,1] => [[4,1],[]] => [[2,1,1,1],[]]
[3,2] => [[3,2],[]] => [[2,2,1],[]]
[3,1,1] => [[3,1,1],[]] => [[3,1,1],[]]
[2,2,1] => [[2,2,1],[]] => [[3,2],[]]
[2,1,1,1] => [[2,1,1,1],[]] => [[4,1],[]]
[1,1,1,1,1] => [[1,1,1,1,1],[]] => [[5],[]]
[6] => [[6],[]] => [[1,1,1,1,1,1],[]]
[5,1] => [[5,1],[]] => [[2,1,1,1,1],[]]
[4,2] => [[4,2],[]] => [[2,2,1,1],[]]
[4,1,1] => [[4,1,1],[]] => [[3,1,1,1],[]]
[3,3] => [[3,3],[]] => [[2,2,2],[]]
[3,2,1] => [[3,2,1],[]] => [[3,2,1],[]]
[3,1,1,1] => [[3,1,1,1],[]] => [[4,1,1],[]]
[2,2,2] => [[2,2,2],[]] => [[3,3],[]]
[2,2,1,1] => [[2,2,1,1],[]] => [[4,2],[]]
[2,1,1,1,1] => [[2,1,1,1,1],[]] => [[5,1],[]]
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => [[6],[]]
[7] => [[7],[]] => [[1,1,1,1,1,1,1],[]]
[6,1] => [[6,1],[]] => [[2,1,1,1,1,1],[]]
[5,2] => [[5,2],[]] => [[2,2,1,1,1],[]]
[5,1,1] => [[5,1,1],[]] => [[3,1,1,1,1],[]]
[4,3] => [[4,3],[]] => [[2,2,2,1],[]]
[4,2,1] => [[4,2,1],[]] => [[3,2,1,1],[]]
[4,1,1,1] => [[4,1,1,1],[]] => [[4,1,1,1],[]]
[3,3,1] => [[3,3,1],[]] => [[3,2,2],[]]
[3,2,2] => [[3,2,2],[]] => [[3,3,1],[]]
[3,2,1,1] => [[3,2,1,1],[]] => [[4,2,1],[]]
[3,1,1,1,1] => [[3,1,1,1,1],[]] => [[5,1,1],[]]
[2,2,2,1] => [[2,2,2,1],[]] => [[4,3],[]]
[2,2,1,1,1] => [[2,2,1,1,1],[]] => [[5,2],[]]
[2,1,1,1,1,1] => [[2,1,1,1,1,1],[]] => [[6,1],[]]
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => [[7],[]]
[8] => [[8],[]] => [[1,1,1,1,1,1,1,1],[]]
[7,1] => [[7,1],[]] => [[2,1,1,1,1,1,1],[]]
[6,2] => [[6,2],[]] => [[2,2,1,1,1,1],[]]
[6,1,1] => [[6,1,1],[]] => [[3,1,1,1,1,1],[]]
[5,3] => [[5,3],[]] => [[2,2,2,1,1],[]]
[5,2,1] => [[5,2,1],[]] => [[3,2,1,1,1],[]]
[5,1,1,1] => [[5,1,1,1],[]] => [[4,1,1,1,1],[]]
[4,4] => [[4,4],[]] => [[2,2,2,2],[]]
[4,3,1] => [[4,3,1],[]] => [[3,2,2,1],[]]
[4,2,2] => [[4,2,2],[]] => [[3,3,1,1],[]]
[4,2,1,1] => [[4,2,1,1],[]] => [[4,2,1,1],[]]
[4,1,1,1,1] => [[4,1,1,1,1],[]] => [[5,1,1,1],[]]
[3,3,2] => [[3,3,2],[]] => [[3,3,2],[]]
[3,3,1,1] => [[3,3,1,1],[]] => [[4,2,2],[]]
[3,2,2,1] => [[3,2,2,1],[]] => [[4,3,1],[]]
[3,2,1,1,1] => [[3,2,1,1,1],[]] => [[5,2,1],[]]
[3,1,1,1,1,1] => [[3,1,1,1,1,1],[]] => [[6,1,1],[]]
[2,2,2,2] => [[2,2,2,2],[]] => [[4,4],[]]
[2,2,2,1,1] => [[2,2,2,1,1],[]] => [[5,3],[]]
[2,2,1,1,1,1] => [[2,2,1,1,1,1],[]] => [[6,2],[]]
[2,1,1,1,1,1,1] => [[2,1,1,1,1,1,1],[]] => [[7,1],[]]
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]] => [[8],[]]
[9] => [[9],[]] => [[1,1,1,1,1,1,1,1,1],[]]
[8,1] => [[8,1],[]] => [[2,1,1,1,1,1,1,1],[]]
[7,2] => [[7,2],[]] => [[2,2,1,1,1,1,1],[]]
[7,1,1] => [[7,1,1],[]] => [[3,1,1,1,1,1,1],[]]
[6,3] => [[6,3],[]] => [[2,2,2,1,1,1],[]]
[6,2,1] => [[6,2,1],[]] => [[3,2,1,1,1,1],[]]
[6,1,1,1] => [[6,1,1,1],[]] => [[4,1,1,1,1,1],[]]
[5,4] => [[5,4],[]] => [[2,2,2,2,1],[]]
[5,3,1] => [[5,3,1],[]] => [[3,2,2,1,1],[]]
[5,2,2] => [[5,2,2],[]] => [[3,3,1,1,1],[]]
[5,2,1,1] => [[5,2,1,1],[]] => [[4,2,1,1,1],[]]
[5,1,1,1,1] => [[5,1,1,1,1],[]] => [[5,1,1,1,1],[]]
[4,4,1] => [[4,4,1],[]] => [[3,2,2,2],[]]
[4,3,2] => [[4,3,2],[]] => [[3,3,2,1],[]]
[4,3,1,1] => [[4,3,1,1],[]] => [[4,2,2,1],[]]
[4,2,2,1] => [[4,2,2,1],[]] => [[4,3,1,1],[]]
[4,2,1,1,1] => [[4,2,1,1,1],[]] => [[5,2,1,1],[]]
[4,1,1,1,1,1] => [[4,1,1,1,1,1],[]] => [[6,1,1,1],[]]
[3,3,3] => [[3,3,3],[]] => [[3,3,3],[]]
[3,3,2,1] => [[3,3,2,1],[]] => [[4,3,2],[]]
[3,3,1,1,1] => [[3,3,1,1,1],[]] => [[5,2,2],[]]
[3,2,2,2] => [[3,2,2,2],[]] => [[4,4,1],[]]
[3,2,2,1,1] => [[3,2,2,1,1],[]] => [[5,3,1],[]]
[3,2,1,1,1,1] => [[3,2,1,1,1,1],[]] => [[6,2,1],[]]
[3,1,1,1,1,1,1] => [[3,1,1,1,1,1,1],[]] => [[7,1,1],[]]
[2,2,2,2,1] => [[2,2,2,2,1],[]] => [[5,4],[]]
[2,2,2,1,1,1] => [[2,2,2,1,1,1],[]] => [[6,3],[]]
[2,2,1,1,1,1,1] => [[2,2,1,1,1,1,1],[]] => [[7,2],[]]
[2,1,1,1,1,1,1,1] => [[2,1,1,1,1,1,1,1],[]] => [[8,1],[]]
[1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1],[]] => [[9],[]]
[10] => [[10],[]] => [[1,1,1,1,1,1,1,1,1,1],[]]
[9,1] => [[9,1],[]] => [[2,1,1,1,1,1,1,1,1],[]]
[8,2] => [[8,2],[]] => [[2,2,1,1,1,1,1,1],[]]
[8,1,1] => [[8,1,1],[]] => [[3,1,1,1,1,1,1,1],[]]
[7,3] => [[7,3],[]] => [[2,2,2,1,1,1,1],[]]
>>> Load all 250 entries. <<<
[7,2,1] => [[7,2,1],[]] => [[3,2,1,1,1,1,1],[]]
[7,1,1,1] => [[7,1,1,1],[]] => [[4,1,1,1,1,1,1],[]]
[6,4] => [[6,4],[]] => [[2,2,2,2,1,1],[]]
[6,3,1] => [[6,3,1],[]] => [[3,2,2,1,1,1],[]]
[6,2,2] => [[6,2,2],[]] => [[3,3,1,1,1,1],[]]
[6,2,1,1] => [[6,2,1,1],[]] => [[4,2,1,1,1,1],[]]
[6,1,1,1,1] => [[6,1,1,1,1],[]] => [[5,1,1,1,1,1],[]]
[5,5] => [[5,5],[]] => [[2,2,2,2,2],[]]
[5,4,1] => [[5,4,1],[]] => [[3,2,2,2,1],[]]
[5,3,2] => [[5,3,2],[]] => [[3,3,2,1,1],[]]
[5,3,1,1] => [[5,3,1,1],[]] => [[4,2,2,1,1],[]]
[5,2,2,1] => [[5,2,2,1],[]] => [[4,3,1,1,1],[]]
[5,2,1,1,1] => [[5,2,1,1,1],[]] => [[5,2,1,1,1],[]]
[5,1,1,1,1,1] => [[5,1,1,1,1,1],[]] => [[6,1,1,1,1],[]]
[4,4,2] => [[4,4,2],[]] => [[3,3,2,2],[]]
[4,4,1,1] => [[4,4,1,1],[]] => [[4,2,2,2],[]]
[4,3,3] => [[4,3,3],[]] => [[3,3,3,1],[]]
[4,3,2,1] => [[4,3,2,1],[]] => [[4,3,2,1],[]]
[4,3,1,1,1] => [[4,3,1,1,1],[]] => [[5,2,2,1],[]]
[4,2,2,2] => [[4,2,2,2],[]] => [[4,4,1,1],[]]
[4,2,2,1,1] => [[4,2,2,1,1],[]] => [[5,3,1,1],[]]
[4,2,1,1,1,1] => [[4,2,1,1,1,1],[]] => [[6,2,1,1],[]]
[4,1,1,1,1,1,1] => [[4,1,1,1,1,1,1],[]] => [[7,1,1,1],[]]
[3,3,3,1] => [[3,3,3,1],[]] => [[4,3,3],[]]
[3,3,2,2] => [[3,3,2,2],[]] => [[4,4,2],[]]
[3,3,2,1,1] => [[3,3,2,1,1],[]] => [[5,3,2],[]]
[3,3,1,1,1,1] => [[3,3,1,1,1,1],[]] => [[6,2,2],[]]
[3,2,2,2,1] => [[3,2,2,2,1],[]] => [[5,4,1],[]]
[3,2,2,1,1,1] => [[3,2,2,1,1,1],[]] => [[6,3,1],[]]
[3,2,1,1,1,1,1] => [[3,2,1,1,1,1,1],[]] => [[7,2,1],[]]
[3,1,1,1,1,1,1,1] => [[3,1,1,1,1,1,1,1],[]] => [[8,1,1],[]]
[2,2,2,2,2] => [[2,2,2,2,2],[]] => [[5,5],[]]
[2,2,2,2,1,1] => [[2,2,2,2,1,1],[]] => [[6,4],[]]
[2,2,2,1,1,1,1] => [[2,2,2,1,1,1,1],[]] => [[7,3],[]]
[2,2,1,1,1,1,1,1] => [[2,2,1,1,1,1,1,1],[]] => [[8,2],[]]
[2,1,1,1,1,1,1,1,1] => [[2,1,1,1,1,1,1,1,1],[]] => [[9,1],[]]
[1,1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1,1],[]] => [[10],[]]
[11] => [[11],[]] => [[1,1,1,1,1,1,1,1,1,1,1],[]]
[9,2] => [[9,2],[]] => [[2,2,1,1,1,1,1,1,1],[]]
[7,4] => [[7,4],[]] => [[2,2,2,2,1,1,1],[]]
[7,2,2] => [[7,2,2],[]] => [[3,3,1,1,1,1,1],[]]
[6,5] => [[6,5],[]] => [[2,2,2,2,2,1],[]]
[5,5,1] => [[5,5,1],[]] => [[3,2,2,2,2],[]]
[5,4,2] => [[5,4,2],[]] => [[3,3,2,2,1],[]]
[5,3,3] => [[5,3,3],[]] => [[3,3,3,1,1],[]]
[5,2,2,2] => [[5,2,2,2],[]] => [[4,4,1,1,1],[]]
[4,4,3] => [[4,4,3],[]] => [[3,3,3,2],[]]
[4,4,2,1] => [[4,4,2,1],[]] => [[4,3,2,2],[]]
[4,3,3,1] => [[4,3,3,1],[]] => [[4,3,3,1],[]]
[4,3,2,2] => [[4,3,2,2],[]] => [[4,4,2,1],[]]
[3,3,3,2] => [[3,3,3,2],[]] => [[4,4,3],[]]
[3,3,3,1,1] => [[3,3,3,1,1],[]] => [[5,3,3],[]]
[3,3,2,2,1] => [[3,3,2,2,1],[]] => [[5,4,2],[]]
[3,2,2,2,2] => [[3,2,2,2,2],[]] => [[5,5,1],[]]
[3,2,2,2,1,1] => [[3,2,2,2,1,1],[]] => [[6,4,1],[]]
[3,1,1,1,1,1,1,1,1] => [[3,1,1,1,1,1,1,1,1],[]] => [[9,1,1],[]]
[2,2,2,2,2,1] => [[2,2,2,2,2,1],[]] => [[6,5],[]]
[2,1,1,1,1,1,1,1,1,1] => [[2,1,1,1,1,1,1,1,1,1],[]] => [[10,1],[]]
[7,5] => [[7,5],[]] => [[2,2,2,2,2,1,1],[]]
[6,6] => [[6,6],[]] => [[2,2,2,2,2,2],[]]
[6,5,1] => [[6,5,1],[]] => [[3,2,2,2,2,1],[]]
[5,5,2] => [[5,5,2],[]] => [[3,3,2,2,2],[]]
[5,5,1,1] => [[5,5,1,1],[]] => [[4,2,2,2,2],[]]
[5,4,3] => [[5,4,3],[]] => [[3,3,3,2,1],[]]
[4,4,4] => [[4,4,4],[]] => [[3,3,3,3],[]]
[4,4,3,1] => [[4,4,3,1],[]] => [[4,3,3,2],[]]
[4,4,2,2] => [[4,4,2,2],[]] => [[4,4,2,2],[]]
[4,3,3,2] => [[4,3,3,2],[]] => [[4,4,3,1],[]]
[4,2,2,2,1,1] => [[4,2,2,2,1,1],[]] => [[6,4,1,1],[]]
[3,3,3,3] => [[3,3,3,3],[]] => [[4,4,4],[]]
[3,3,3,2,1] => [[3,3,3,2,1],[]] => [[5,4,3],[]]
[3,3,3,1,1,1] => [[3,3,3,1,1,1],[]] => [[6,3,3],[]]
[3,3,2,2,2] => [[3,3,2,2,2],[]] => [[5,5,2],[]]
[3,2,2,2,2,1] => [[3,2,2,2,2,1],[]] => [[6,5,1],[]]
[2,2,2,2,2,2] => [[2,2,2,2,2,2],[]] => [[6,6],[]]
[7,6] => [[7,6],[]] => [[2,2,2,2,2,2,1],[]]
[6,6,1] => [[6,6,1],[]] => [[3,2,2,2,2,2],[]]
[5,5,3] => [[5,5,3],[]] => [[3,3,3,2,2],[]]
[5,4,4] => [[5,4,4],[]] => [[3,3,3,3,1],[]]
[4,4,4,1] => [[4,4,4,1],[]] => [[4,3,3,3],[]]
[4,4,3,2] => [[4,4,3,2],[]] => [[4,4,3,2],[]]
[4,4,3,1,1] => [[4,4,3,1,1],[]] => [[5,3,3,2],[]]
[4,3,3,3] => [[4,3,3,3],[]] => [[4,4,4,1],[]]
[3,3,3,3,1] => [[3,3,3,3,1],[]] => [[5,4,4],[]]
[3,3,3,2,2] => [[3,3,3,2,2],[]] => [[5,5,3],[]]
[3,2,2,2,2,2] => [[3,2,2,2,2,2],[]] => [[6,6,1],[]]
[2,2,2,2,2,2,1] => [[2,2,2,2,2,2,1],[]] => [[7,6],[]]
[7,7] => [[7,7],[]] => [[2,2,2,2,2,2,2],[]]
[6,6,2] => [[6,6,2],[]] => [[3,3,2,2,2,2],[]]
[6,4,4] => [[6,4,4],[]] => [[3,3,3,3,1,1],[]]
[6,2,2,2,2] => [[6,2,2,2,2],[]] => [[5,5,1,1,1,1],[]]
[5,5,4] => [[5,5,4],[]] => [[3,3,3,3,2],[]]
[5,5,3,1] => [[5,5,3,1],[]] => [[4,3,3,2,2],[]]
[4,4,4,2] => [[4,4,4,2],[]] => [[4,4,3,3],[]]
[4,4,4,1,1] => [[4,4,4,1,1],[]] => [[5,3,3,3],[]]
[4,4,3,3] => [[4,4,3,3],[]] => [[4,4,4,2],[]]
[3,3,3,3,2] => [[3,3,3,3,2],[]] => [[5,5,4],[]]
[3,3,3,3,1,1] => [[3,3,3,3,1,1],[]] => [[6,4,4],[]]
[3,3,2,2,2,2] => [[3,3,2,2,2,2],[]] => [[6,6,2],[]]
[3,2,2,2,2,2,1] => [[3,2,2,2,2,2,1],[]] => [[7,6,1],[]]
[2,2,2,2,2,2,2] => [[2,2,2,2,2,2,2],[]] => [[7,7],[]]
[7,7,1] => [[7,7,1],[]] => [[3,2,2,2,2,2,2],[]]
[6,6,3] => [[6,6,3],[]] => [[3,3,3,2,2,2],[]]
[6,5,4] => [[6,5,4],[]] => [[3,3,3,3,2,1],[]]
[5,5,5] => [[5,5,5],[]] => [[3,3,3,3,3],[]]
[5,5,4,1] => [[5,5,4,1],[]] => [[4,3,3,3,2],[]]
[4,4,4,3] => [[4,4,4,3],[]] => [[4,4,4,3],[]]
[4,4,3,3,1] => [[4,4,3,3,1],[]] => [[5,4,4,2],[]]
[3,3,3,3,3] => [[3,3,3,3,3],[]] => [[5,5,5],[]]
[3,2,2,2,2,2,2] => [[3,2,2,2,2,2,2],[]] => [[7,7,1],[]]
[2,2,2,2,2,2,2,1] => [[2,2,2,2,2,2,2,1],[]] => [[8,7],[]]
[8,8] => [[8,8],[]] => [[2,2,2,2,2,2,2,2],[]]
[6,6,3,1] => [[6,6,3,1],[]] => [[4,3,3,2,2,2],[]]
[6,5,5] => [[6,5,5],[]] => [[3,3,3,3,3,1],[]]
[5,5,5,1] => [[5,5,5,1],[]] => [[4,3,3,3,3],[]]
[5,5,3,3] => [[5,5,3,3],[]] => [[4,4,4,2,2],[]]
[5,4,4,3] => [[5,4,4,3],[]] => [[4,4,4,3,1],[]]
[4,4,4,4] => [[4,4,4,4],[]] => [[4,4,4,4],[]]
[4,4,4,3,1] => [[4,4,4,3,1],[]] => [[5,4,4,3],[]]
[4,3,3,3,3] => [[4,3,3,3,3],[]] => [[5,5,5,1],[]]
[3,3,3,3,3,1] => [[3,3,3,3,3,1],[]] => [[6,5,5],[]]
[2,2,2,2,2,2,2,2] => [[2,2,2,2,2,2,2,2],[]] => [[8,8],[]]
[8,8,1] => [[8,8,1],[]] => [[3,2,2,2,2,2,2,2],[]]
[7,7,3] => [[7,7,3],[]] => [[3,3,3,2,2,2,2],[]]
[5,5,5,2] => [[5,5,5,2],[]] => [[4,4,3,3,3],[]]
[5,4,4,4] => [[5,4,4,4],[]] => [[4,4,4,4,1],[]]
[4,4,4,4,1] => [[4,4,4,4,1],[]] => [[5,4,4,4],[]]
[4,4,4,3,2] => [[4,4,4,3,2],[]] => [[5,5,4,3],[]]
[3,2,2,2,2,2,2,2] => [[3,2,2,2,2,2,2,2],[]] => [[8,8,1],[]]
[2,2,2,2,2,2,2,2,1] => [[2,2,2,2,2,2,2,2,1],[]] => [[9,8],[]]
[4,4,4,4,2] => [[4,4,4,4,2],[]] => [[5,5,4,4],[]]
[6,5,5,4] => [[6,5,5,4],[]] => [[4,4,4,4,3,1],[]]
[3,3,3,3,3,3] => [[3,3,3,3,3,3],[]] => [[6,6,6],[]]
[3,3,3,3,3,3,1] => [[3,3,3,3,3,3,1],[]] => [[7,6,6],[]]
[4,4,4,4,4] => [[4,4,4,4,4],[]] => [[5,5,5,5],[]]
[4,4,4,3,3,3] => [[4,4,4,3,3,3],[]] => [[6,6,6,3],[]]
[4,4,4,4,4,4] => [[4,4,4,4,4,4],[]] => [[6,6,6,6],[]]
[5,5,5,5] => [[5,5,5,5],[]] => [[4,4,4,4,4],[]]
[6,6,6,6] => [[6,6,6,6],[]] => [[4,4,4,4,4,4],[]]
[6,6,6] => [[6,6,6],[]] => [[3,3,3,3,3,3],[]]
[6,6,6,6,6] => [[6,6,6,6,6],[]] => [[5,5,5,5,5,5],[]]
[5,5,5,5,5,5] => [[5,5,5,5,5,5],[]] => [[6,6,6,6,6],[]]
[5,5,5,5,5] => [[5,5,5,5,5],[]] => [[5,5,5,5,5],[]]
[9,9] => [[9,9],[]] => [[2,2,2,2,2,2,2,2,2],[]]
[6,5,5,5] => [[6,5,5,5],[]] => [[4,4,4,4,4,1],[]]
[5,4,4,4,4] => [[5,4,4,4,4],[]] => [[5,5,5,5,1],[]]
[4,4,4,4,4,1] => [[4,4,4,4,4,1],[]] => [[6,5,5,5],[]]
[5,5,5,5,1] => [[5,5,5,5,1],[]] => [[5,4,4,4,4],[]]
[6,6,6,1] => [[6,6,6,1],[]] => [[4,3,3,3,3,3],[]]
Map
to skew partition
Description
The partition regarded as a skew partition.
Map
conjugate
Description
The conjugate of the skew partition.
The conjugate of a skew partition $\lambda$ is the skew partition $\lambda^*$ whose Ferrers diagram is obtained from the Ferrers diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.