Identifier
-
Mp00179:
Integer partitions
—to skew partition⟶
Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001924: Integer partitions ⟶ ℤ
Values
[1] => [[1],[]] => [[1],[]] => [1] => 1
[2] => [[2],[]] => [[2],[]] => [2] => 1
[1,1] => [[1,1],[]] => [[1,1],[]] => [1,1] => 1
[3] => [[3],[]] => [[3],[]] => [3] => 1
[2,1] => [[2,1],[]] => [[2,2],[1]] => [2,2] => 2
[1,1,1] => [[1,1,1],[]] => [[1,1,1],[]] => [1,1,1] => 1
[4] => [[4],[]] => [[4],[]] => [4] => 1
[3,1] => [[3,1],[]] => [[3,3],[2]] => [3,3] => 2
[2,2] => [[2,2],[]] => [[2,2],[]] => [2,2] => 2
[2,1,1] => [[2,1,1],[]] => [[2,2,2],[1,1]] => [2,2,2] => 2
[1,1,1,1] => [[1,1,1,1],[]] => [[1,1,1,1],[]] => [1,1,1,1] => 1
[5] => [[5],[]] => [[5],[]] => [5] => 1
[4,1] => [[4,1],[]] => [[4,4],[3]] => [4,4] => 2
[3,2] => [[3,2],[]] => [[3,3],[1]] => [3,3] => 2
[3,1,1] => [[3,1,1],[]] => [[3,3,3],[2,2]] => [3,3,3] => 3
[2,2,1] => [[2,2,1],[]] => [[2,2,2],[1]] => [2,2,2] => 2
[2,1,1,1] => [[2,1,1,1],[]] => [[2,2,2,2],[1,1,1]] => [2,2,2,2] => 2
[1,1,1,1,1] => [[1,1,1,1,1],[]] => [[1,1,1,1,1],[]] => [1,1,1,1,1] => 1
[6] => [[6],[]] => [[6],[]] => [6] => 1
[5,1] => [[5,1],[]] => [[5,5],[4]] => [5,5] => 2
[4,2] => [[4,2],[]] => [[4,4],[2]] => [4,4] => 2
[4,1,1] => [[4,1,1],[]] => [[4,4,4],[3,3]] => [4,4,4] => 3
[3,3] => [[3,3],[]] => [[3,3],[]] => [3,3] => 2
[3,2,1] => [[3,2,1],[]] => [[3,3,3],[2,1]] => [3,3,3] => 3
[3,1,1,1] => [[3,1,1,1],[]] => [[3,3,3,3],[2,2,2]] => [3,3,3,3] => 3
[2,2,2] => [[2,2,2],[]] => [[2,2,2],[]] => [2,2,2] => 2
[2,2,1,1] => [[2,2,1,1],[]] => [[2,2,2,2],[1,1]] => [2,2,2,2] => 2
[2,1,1,1,1] => [[2,1,1,1,1],[]] => [[2,2,2,2,2],[1,1,1,1]] => [2,2,2,2,2] => 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
[7] => [[7],[]] => [[7],[]] => [7] => 1
[6,1] => [[6,1],[]] => [[6,6],[5]] => [6,6] => 2
[5,2] => [[5,2],[]] => [[5,5],[3]] => [5,5] => 2
[5,1,1] => [[5,1,1],[]] => [[5,5,5],[4,4]] => [5,5,5] => 3
[4,3] => [[4,3],[]] => [[4,4],[1]] => [4,4] => 2
[4,2,1] => [[4,2,1],[]] => [[4,4,4],[3,2]] => [4,4,4] => 3
[4,1,1,1] => [[4,1,1,1],[]] => [[4,4,4,4],[3,3,3]] => [4,4,4,4] => 4
[3,3,1] => [[3,3,1],[]] => [[3,3,3],[2]] => [3,3,3] => 3
[3,2,2] => [[3,2,2],[]] => [[3,3,3],[1,1]] => [3,3,3] => 3
[3,2,1,1] => [[3,2,1,1],[]] => [[3,3,3,3],[2,2,1]] => [3,3,3,3] => 3
[3,1,1,1,1] => [[3,1,1,1,1],[]] => [[3,3,3,3,3],[2,2,2,2]] => [3,3,3,3,3] => 3
[2,2,2,1] => [[2,2,2,1],[]] => [[2,2,2,2],[1]] => [2,2,2,2] => 2
[2,2,1,1,1] => [[2,2,1,1,1],[]] => [[2,2,2,2,2],[1,1,1]] => [2,2,2,2,2] => 2
[2,1,1,1,1,1] => [[2,1,1,1,1,1],[]] => [[2,2,2,2,2,2],[1,1,1,1,1]] => [2,2,2,2,2,2] => 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
[8] => [[8],[]] => [[8],[]] => [8] => 1
[7,1] => [[7,1],[]] => [[7,7],[6]] => [7,7] => 2
[6,2] => [[6,2],[]] => [[6,6],[4]] => [6,6] => 2
[5,3] => [[5,3],[]] => [[5,5],[2]] => [5,5] => 2
[5,2,1] => [[5,2,1],[]] => [[5,5,5],[4,3]] => [5,5,5] => 3
[4,4] => [[4,4],[]] => [[4,4],[]] => [4,4] => 2
[4,3,1] => [[4,3,1],[]] => [[4,4,4],[3,1]] => [4,4,4] => 3
[4,2,2] => [[4,2,2],[]] => [[4,4,4],[2,2]] => [4,4,4] => 3
[4,2,1,1] => [[4,2,1,1],[]] => [[4,4,4,4],[3,3,2]] => [4,4,4,4] => 4
[3,3,2] => [[3,3,2],[]] => [[3,3,3],[1]] => [3,3,3] => 3
[3,3,1,1] => [[3,3,1,1],[]] => [[3,3,3,3],[2,2]] => [3,3,3,3] => 3
[3,2,2,1] => [[3,2,2,1],[]] => [[3,3,3,3],[2,1,1]] => [3,3,3,3] => 3
[3,2,1,1,1] => [[3,2,1,1,1],[]] => [[3,3,3,3,3],[2,2,2,1]] => [3,3,3,3,3] => 3
[2,2,2,2] => [[2,2,2,2],[]] => [[2,2,2,2],[]] => [2,2,2,2] => 2
[2,2,2,1,1] => [[2,2,2,1,1],[]] => [[2,2,2,2,2],[1,1]] => [2,2,2,2,2] => 2
[2,2,1,1,1,1] => [[2,2,1,1,1,1],[]] => [[2,2,2,2,2,2],[1,1,1,1]] => [2,2,2,2,2,2] => 2
[2,1,1,1,1,1,1] => [[2,1,1,1,1,1,1],[]] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]] => [2,2,2,2,2,2,2] => 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
[9] => [[9],[]] => [[9],[]] => [9] => 1
[8,1] => [[8,1],[]] => [[8,8],[7]] => [8,8] => 2
[7,2] => [[7,2],[]] => [[7,7],[5]] => [7,7] => 2
[6,3] => [[6,3],[]] => [[6,6],[3]] => [6,6] => 2
[5,4] => [[5,4],[]] => [[5,5],[1]] => [5,5] => 2
[5,3,1] => [[5,3,1],[]] => [[5,5,5],[4,2]] => [5,5,5] => 3
[5,2,2] => [[5,2,2],[]] => [[5,5,5],[3,3]] => [5,5,5] => 3
[4,4,1] => [[4,4,1],[]] => [[4,4,4],[3]] => [4,4,4] => 3
[4,3,2] => [[4,3,2],[]] => [[4,4,4],[2,1]] => [4,4,4] => 3
[4,3,1,1] => [[4,3,1,1],[]] => [[4,4,4,4],[3,3,1]] => [4,4,4,4] => 4
[4,2,2,1] => [[4,2,2,1],[]] => [[4,4,4,4],[3,2,2]] => [4,4,4,4] => 4
[3,3,3] => [[3,3,3],[]] => [[3,3,3],[]] => [3,3,3] => 3
[3,3,2,1] => [[3,3,2,1],[]] => [[3,3,3,3],[2,1]] => [3,3,3,3] => 3
[3,3,1,1,1] => [[3,3,1,1,1],[]] => [[3,3,3,3,3],[2,2,2]] => [3,3,3,3,3] => 3
[3,2,2,2] => [[3,2,2,2],[]] => [[3,3,3,3],[1,1,1]] => [3,3,3,3] => 3
[3,2,2,1,1] => [[3,2,2,1,1],[]] => [[3,3,3,3,3],[2,2,1,1]] => [3,3,3,3,3] => 3
[2,2,2,2,1] => [[2,2,2,2,1],[]] => [[2,2,2,2,2],[1]] => [2,2,2,2,2] => 2
[2,2,2,1,1,1] => [[2,2,2,1,1,1],[]] => [[2,2,2,2,2,2],[1,1,1]] => [2,2,2,2,2,2] => 2
[2,2,1,1,1,1,1] => [[2,2,1,1,1,1,1],[]] => [[2,2,2,2,2,2,2],[1,1,1,1,1]] => [2,2,2,2,2,2,2] => 2
[2,1,1,1,1,1,1,1] => [[2,1,1,1,1,1,1,1],[]] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]] => [2,2,2,2,2,2,2,2] => 2
[1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1],[]] => [[1,1,1,1,1,1,1,1,1],[]] => [1,1,1,1,1,1,1,1,1] => 1
[10] => [[10],[]] => [[10],[]] => [10] => 1
[8,2] => [[8,2],[]] => [[8,8],[6]] => [8,8] => 2
[7,3] => [[7,3],[]] => [[7,7],[4]] => [7,7] => 2
[6,4] => [[6,4],[]] => [[6,6],[2]] => [6,6] => 2
[5,5] => [[5,5],[]] => [[5,5],[]] => [5,5] => 2
[5,4,1] => [[5,4,1],[]] => [[5,5,5],[4,1]] => [5,5,5] => 3
[5,3,2] => [[5,3,2],[]] => [[5,5,5],[3,2]] => [5,5,5] => 3
[4,4,2] => [[4,4,2],[]] => [[4,4,4],[2]] => [4,4,4] => 3
[4,4,1,1] => [[4,4,1,1],[]] => [[4,4,4,4],[3,3]] => [4,4,4,4] => 4
[4,3,3] => [[4,3,3],[]] => [[4,4,4],[1,1]] => [4,4,4] => 3
[4,3,2,1] => [[4,3,2,1],[]] => [[4,4,4,4],[3,2,1]] => [4,4,4,4] => 4
[4,2,2,2] => [[4,2,2,2],[]] => [[4,4,4,4],[2,2,2]] => [4,4,4,4] => 4
[3,3,3,1] => [[3,3,3,1],[]] => [[3,3,3,3],[2]] => [3,3,3,3] => 3
[3,3,2,2] => [[3,3,2,2],[]] => [[3,3,3,3],[1,1]] => [3,3,3,3] => 3
[3,3,2,1,1] => [[3,3,2,1,1],[]] => [[3,3,3,3,3],[2,2,1]] => [3,3,3,3,3] => 3
[3,2,2,2,1] => [[3,2,2,2,1],[]] => [[3,3,3,3,3],[2,1,1,1]] => [3,3,3,3,3] => 3
[2,2,2,2,2] => [[2,2,2,2,2],[]] => [[2,2,2,2,2],[]] => [2,2,2,2,2] => 2
[2,2,2,2,1,1] => [[2,2,2,2,1,1],[]] => [[2,2,2,2,2,2],[1,1]] => [2,2,2,2,2,2] => 2
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Description
The number of cells in an integer partition whose arm and leg length coincide.
Map
outer shape
Description
The outer shape of the skew partition.
Map
to skew partition
Description
The partition regarded as a skew partition.
Map
rotate
Description
The rotation of a skew partition.
This is the skew partition obtained by rotating the diagram by 180 degrees. Equivalently, given a skew partition $\lambda/\mu$, its rotation $(\lambda/\mu)^\natural$ is the skew partition with cells $\{(a-i, b-j)| (i, j) \in \lambda/\mu\}$, where $b$ and $a$ are the first part and the number of parts of $\lambda$ respectively.
This is the skew partition obtained by rotating the diagram by 180 degrees. Equivalently, given a skew partition $\lambda/\mu$, its rotation $(\lambda/\mu)^\natural$ is the skew partition with cells $\{(a-i, b-j)| (i, j) \in \lambda/\mu\}$, where $b$ and $a$ are the first part and the number of parts of $\lambda$ respectively.
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