Identifier
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00186: Skew partitions —dominating partition⟶ Integer partitions
Images
[1] => [[1],[]] => [1]
[2] => [[2],[]] => [2]
[1,1] => [[1,1],[]] => [1,1]
[3] => [[3],[]] => [3]
[2,1] => [[2,1],[]] => [2,1]
[1,1,1] => [[1,1,1],[]] => [1,1,1]
[4] => [[4],[]] => [4]
[3,1] => [[3,1],[]] => [3,1]
[2,2] => [[2,2],[]] => [2,2]
[2,1,1] => [[2,1,1],[]] => [2,1,1]
[1,1,1,1] => [[1,1,1,1],[]] => [1,1,1,1]
[5] => [[5],[]] => [5]
[4,1] => [[4,1],[]] => [4,1]
[3,2] => [[3,2],[]] => [3,2]
[3,1,1] => [[3,1,1],[]] => [3,1,1]
[2,2,1] => [[2,2,1],[]] => [2,2,1]
[2,1,1,1] => [[2,1,1,1],[]] => [2,1,1,1]
[1,1,1,1,1] => [[1,1,1,1,1],[]] => [1,1,1,1,1]
[6] => [[6],[]] => [6]
[5,1] => [[5,1],[]] => [5,1]
[4,2] => [[4,2],[]] => [4,2]
[4,1,1] => [[4,1,1],[]] => [4,1,1]
[3,3] => [[3,3],[]] => [3,3]
[3,2,1] => [[3,2,1],[]] => [3,2,1]
[3,1,1,1] => [[3,1,1,1],[]] => [3,1,1,1]
[2,2,2] => [[2,2,2],[]] => [2,2,2]
[2,2,1,1] => [[2,2,1,1],[]] => [2,2,1,1]
[2,1,1,1,1] => [[2,1,1,1,1],[]] => [2,1,1,1,1]
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => [1,1,1,1,1,1]
[7] => [[7],[]] => [7]
[6,1] => [[6,1],[]] => [6,1]
[5,2] => [[5,2],[]] => [5,2]
[5,1,1] => [[5,1,1],[]] => [5,1,1]
[4,3] => [[4,3],[]] => [4,3]
[4,2,1] => [[4,2,1],[]] => [4,2,1]
[4,1,1,1] => [[4,1,1,1],[]] => [4,1,1,1]
[3,3,1] => [[3,3,1],[]] => [3,3,1]
[3,2,2] => [[3,2,2],[]] => [3,2,2]
[3,2,1,1] => [[3,2,1,1],[]] => [3,2,1,1]
[3,1,1,1,1] => [[3,1,1,1,1],[]] => [3,1,1,1,1]
[2,2,2,1] => [[2,2,2,1],[]] => [2,2,2,1]
[2,2,1,1,1] => [[2,2,1,1,1],[]] => [2,2,1,1,1]
[2,1,1,1,1,1] => [[2,1,1,1,1,1],[]] => [2,1,1,1,1,1]
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1]
[8] => [[8],[]] => [8]
[7,1] => [[7,1],[]] => [7,1]
[6,2] => [[6,2],[]] => [6,2]
[6,1,1] => [[6,1,1],[]] => [6,1,1]
[5,3] => [[5,3],[]] => [5,3]
[5,2,1] => [[5,2,1],[]] => [5,2,1]
[5,1,1,1] => [[5,1,1,1],[]] => [5,1,1,1]
[4,4] => [[4,4],[]] => [4,4]
[4,3,1] => [[4,3,1],[]] => [4,3,1]
[4,2,2] => [[4,2,2],[]] => [4,2,2]
[4,2,1,1] => [[4,2,1,1],[]] => [4,2,1,1]
[4,1,1,1,1] => [[4,1,1,1,1],[]] => [4,1,1,1,1]
[3,3,2] => [[3,3,2],[]] => [3,3,2]
[3,3,1,1] => [[3,3,1,1],[]] => [3,3,1,1]
[3,2,2,1] => [[3,2,2,1],[]] => [3,2,2,1]
[3,2,1,1,1] => [[3,2,1,1,1],[]] => [3,2,1,1,1]
[3,1,1,1,1,1] => [[3,1,1,1,1,1],[]] => [3,1,1,1,1,1]
[2,2,2,2] => [[2,2,2,2],[]] => [2,2,2,2]
[2,2,2,1,1] => [[2,2,2,1,1],[]] => [2,2,2,1,1]
[2,2,1,1,1,1] => [[2,2,1,1,1,1],[]] => [2,2,1,1,1,1]
[2,1,1,1,1,1,1] => [[2,1,1,1,1,1,1],[]] => [2,1,1,1,1,1,1]
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1,1]
[9] => [[9],[]] => [9]
[8,1] => [[8,1],[]] => [8,1]
[7,2] => [[7,2],[]] => [7,2]
[7,1,1] => [[7,1,1],[]] => [7,1,1]
[6,3] => [[6,3],[]] => [6,3]
[6,2,1] => [[6,2,1],[]] => [6,2,1]
[6,1,1,1] => [[6,1,1,1],[]] => [6,1,1,1]
[5,4] => [[5,4],[]] => [5,4]
[5,3,1] => [[5,3,1],[]] => [5,3,1]
[5,2,2] => [[5,2,2],[]] => [5,2,2]
[5,2,1,1] => [[5,2,1,1],[]] => [5,2,1,1]
[5,1,1,1,1] => [[5,1,1,1,1],[]] => [5,1,1,1,1]
[4,4,1] => [[4,4,1],[]] => [4,4,1]
[4,3,2] => [[4,3,2],[]] => [4,3,2]
[4,3,1,1] => [[4,3,1,1],[]] => [4,3,1,1]
[4,2,2,1] => [[4,2,2,1],[]] => [4,2,2,1]
[4,2,1,1,1] => [[4,2,1,1,1],[]] => [4,2,1,1,1]
[4,1,1,1,1,1] => [[4,1,1,1,1,1],[]] => [4,1,1,1,1,1]
[3,3,3] => [[3,3,3],[]] => [3,3,3]
[3,3,2,1] => [[3,3,2,1],[]] => [3,3,2,1]
[3,3,1,1,1] => [[3,3,1,1,1],[]] => [3,3,1,1,1]
[3,2,2,2] => [[3,2,2,2],[]] => [3,2,2,2]
[3,2,2,1,1] => [[3,2,2,1,1],[]] => [3,2,2,1,1]
[3,2,1,1,1,1] => [[3,2,1,1,1,1],[]] => [3,2,1,1,1,1]
[3,1,1,1,1,1,1] => [[3,1,1,1,1,1,1],[]] => [3,1,1,1,1,1,1]
[2,2,2,2,1] => [[2,2,2,2,1],[]] => [2,2,2,2,1]
[2,2,2,1,1,1] => [[2,2,2,1,1,1],[]] => [2,2,2,1,1,1]
[2,2,1,1,1,1,1] => [[2,2,1,1,1,1,1],[]] => [2,2,1,1,1,1,1]
[2,1,1,1,1,1,1,1] => [[2,1,1,1,1,1,1,1],[]] => [2,1,1,1,1,1,1,1]
[1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1,1,1]
[10] => [[10],[]] => [10]
[9,1] => [[9,1],[]] => [9,1]
[8,2] => [[8,2],[]] => [8,2]
[8,1,1] => [[8,1,1],[]] => [8,1,1]
[7,3] => [[7,3],[]] => [7,3]
>>> Load all 181 entries. <<<
[7,2,1] => [[7,2,1],[]] => [7,2,1]
[7,1,1,1] => [[7,1,1,1],[]] => [7,1,1,1]
[6,4] => [[6,4],[]] => [6,4]
[6,3,1] => [[6,3,1],[]] => [6,3,1]
[6,2,2] => [[6,2,2],[]] => [6,2,2]
[6,2,1,1] => [[6,2,1,1],[]] => [6,2,1,1]
[6,1,1,1,1] => [[6,1,1,1,1],[]] => [6,1,1,1,1]
[5,5] => [[5,5],[]] => [5,5]
[5,4,1] => [[5,4,1],[]] => [5,4,1]
[5,3,2] => [[5,3,2],[]] => [5,3,2]
[5,3,1,1] => [[5,3,1,1],[]] => [5,3,1,1]
[5,2,2,1] => [[5,2,2,1],[]] => [5,2,2,1]
[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],[]] => [5,1,1,1,1,1]
[4,4,2] => [[4,4,2],[]] => [4,4,2]
[4,4,1,1] => [[4,4,1,1],[]] => [4,4,1,1]
[4,3,3] => [[4,3,3],[]] => [4,3,3]
[4,3,2,1] => [[4,3,2,1],[]] => [4,3,2,1]
[4,3,1,1,1] => [[4,3,1,1,1],[]] => [4,3,1,1,1]
[4,2,2,2] => [[4,2,2,2],[]] => [4,2,2,2]
[4,2,2,1,1] => [[4,2,2,1,1],[]] => [4,2,2,1,1]
[4,2,1,1,1,1] => [[4,2,1,1,1,1],[]] => [4,2,1,1,1,1]
[4,1,1,1,1,1,1] => [[4,1,1,1,1,1,1],[]] => [4,1,1,1,1,1,1]
[3,3,3,1] => [[3,3,3,1],[]] => [3,3,3,1]
[3,3,2,2] => [[3,3,2,2],[]] => [3,3,2,2]
[3,3,2,1,1] => [[3,3,2,1,1],[]] => [3,3,2,1,1]
[3,3,1,1,1,1] => [[3,3,1,1,1,1],[]] => [3,3,1,1,1,1]
[3,2,2,2,1] => [[3,2,2,2,1],[]] => [3,2,2,2,1]
[3,2,2,1,1,1] => [[3,2,2,1,1,1],[]] => [3,2,2,1,1,1]
[3,2,1,1,1,1,1] => [[3,2,1,1,1,1,1],[]] => [3,2,1,1,1,1,1]
[3,1,1,1,1,1,1,1] => [[3,1,1,1,1,1,1,1],[]] => [3,1,1,1,1,1,1,1]
[2,2,2,2,2] => [[2,2,2,2,2],[]] => [2,2,2,2,2]
[2,2,2,2,1,1] => [[2,2,2,2,1,1],[]] => [2,2,2,2,1,1]
[2,2,2,1,1,1,1] => [[2,2,2,1,1,1,1],[]] => [2,2,2,1,1,1,1]
[2,2,1,1,1,1,1,1] => [[2,2,1,1,1,1,1,1],[]] => [2,2,1,1,1,1,1,1]
[2,1,1,1,1,1,1,1,1] => [[2,1,1,1,1,1,1,1,1],[]] => [2,1,1,1,1,1,1,1,1]
[1,1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1,1,1,1]
[6,5] => [[6,5],[]] => [6,5]
[5,5,1] => [[5,5,1],[]] => [5,5,1]
[5,4,2] => [[5,4,2],[]] => [5,4,2]
[5,3,3] => [[5,3,3],[]] => [5,3,3]
[4,4,3] => [[4,4,3],[]] => [4,4,3]
[4,4,2,1] => [[4,4,2,1],[]] => [4,4,2,1]
[4,3,3,1] => [[4,3,3,1],[]] => [4,3,3,1]
[4,3,2,2] => [[4,3,2,2],[]] => [4,3,2,2]
[3,3,3,2] => [[3,3,3,2],[]] => [3,3,3,2]
[3,2,2,2,2] => [[3,2,2,2,2],[]] => [3,2,2,2,2]
[6,6] => [[6,6],[]] => [6,6]
[5,5,2] => [[5,5,2],[]] => [5,5,2]
[5,4,3] => [[5,4,3],[]] => [5,4,3]
[4,4,4] => [[4,4,4],[]] => [4,4,4]
[4,4,3,1] => [[4,4,3,1],[]] => [4,4,3,1]
[4,4,2,2] => [[4,4,2,2],[]] => [4,4,2,2]
[4,3,3,2] => [[4,3,3,2],[]] => [4,3,3,2]
[3,3,3,3] => [[3,3,3,3],[]] => [3,3,3,3]
[3,3,3,2,1] => [[3,3,3,2,1],[]] => [3,3,3,2,1]
[3,3,2,2,2] => [[3,3,2,2,2],[]] => [3,3,2,2,2]
[2,2,2,2,2,2] => [[2,2,2,2,2,2],[]] => [2,2,2,2,2,2]
[5,5,3] => [[5,5,3],[]] => [5,5,3]
[5,4,4] => [[5,4,4],[]] => [5,4,4]
[4,4,4,1] => [[4,4,4,1],[]] => [4,4,4,1]
[4,4,3,2] => [[4,4,3,2],[]] => [4,4,3,2]
[4,3,3,3] => [[4,3,3,3],[]] => [4,3,3,3]
[3,3,3,3,1] => [[3,3,3,3,1],[]] => [3,3,3,3,1]
[3,3,3,2,2] => [[3,3,3,2,2],[]] => [3,3,3,2,2]
[5,5,4] => [[5,5,4],[]] => [5,5,4]
[4,4,4,2] => [[4,4,4,2],[]] => [4,4,4,2]
[4,4,3,3] => [[4,4,3,3],[]] => [4,4,3,3]
[3,3,3,3,2] => [[3,3,3,3,2],[]] => [3,3,3,3,2]
[5,5,5] => [[5,5,5],[]] => [5,5,5]
[5,5,4,1] => [[5,5,4,1],[]] => [5,5,4,1]
[4,4,4,3] => [[4,4,4,3],[]] => [4,4,4,3]
[3,3,3,3,3] => [[3,3,3,3,3],[]] => [3,3,3,3,3]
[5,5,5,1] => [[5,5,5,1],[]] => [5,5,5,1]
[4,4,4,4] => [[4,4,4,4],[]] => [4,4,4,4]
[4,4,4,3,1] => [[4,4,4,3,1],[]] => [4,4,4,3,1]
[3,3,3,3,3,1] => [[3,3,3,3,3,1],[]] => [3,3,3,3,3,1]
[4,4,4,4,1] => [[4,4,4,4,1],[]] => [4,4,4,4,1]
[4,4,4,4,4,1] => [[4,4,4,4,4,1],[]] => [4,4,4,4,4,1]
[5,5,5,5,1] => [[5,5,5,5,1],[]] => [5,5,5,5,1]
Map
to skew partition
Description
The partition regarded as a skew partition.
Map
dominating partition
Description
The dominating partition in the Schur expansion.
Consider the expansion of the skew Schur function $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu, \nu} s_\nu$ as a linear combination of straight Schur functions.
It is shown in [1] that the partitions $\nu$ with $c^\lambda_{\mu, \nu} > 0$ form a sublattice of the dominance order and that its top element is the conjugate of the partition formed by sorting the column lengths of $\lambda / \mu$ into decreasing order.
This map returns the largest partition $\nu$ in dominance order for which $c^\lambda_{\mu, \nu}$ is positive.
For example,
$$ s_{331/2} = s_{311} + s_{32}, $$
and the partition $32$ dominates $311$.