Identifier
-
Mp00179:
Integer partitions
—to skew partition⟶
Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤ
Values
[1] => [[1],[]] => [[1],[]] => [] => 0
[2] => [[2],[]] => [[2],[]] => [] => 0
[1,1] => [[1,1],[]] => [[1,1],[]] => [] => 0
[3] => [[3],[]] => [[3],[]] => [] => 0
[2,1] => [[2,1],[]] => [[2,2],[1]] => [1] => 1
[1,1,1] => [[1,1,1],[]] => [[1,1,1],[]] => [] => 0
[4] => [[4],[]] => [[4],[]] => [] => 0
[3,1] => [[3,1],[]] => [[3,3],[2]] => [2] => 2
[2,2] => [[2,2],[]] => [[2,2],[]] => [] => 0
[2,1,1] => [[2,1,1],[]] => [[2,2,2],[1,1]] => [1,1] => 1
[1,1,1,1] => [[1,1,1,1],[]] => [[1,1,1,1],[]] => [] => 0
[5] => [[5],[]] => [[5],[]] => [] => 0
[4,1] => [[4,1],[]] => [[4,4],[3]] => [3] => 3
[3,2] => [[3,2],[]] => [[3,3],[1]] => [1] => 1
[3,1,1] => [[3,1,1],[]] => [[3,3,3],[2,2]] => [2,2] => 2
[2,2,1] => [[2,2,1],[]] => [[2,2,2],[1]] => [1] => 1
[2,1,1,1] => [[2,1,1,1],[]] => [[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],[]] => [] => 0
[6] => [[6],[]] => [[6],[]] => [] => 0
[5,1] => [[5,1],[]] => [[5,5],[4]] => [4] => 4
[4,2] => [[4,2],[]] => [[4,4],[2]] => [2] => 2
[4,1,1] => [[4,1,1],[]] => [[4,4,4],[3,3]] => [3,3] => 3
[3,3] => [[3,3],[]] => [[3,3],[]] => [] => 0
[3,2,1] => [[3,2,1],[]] => [[3,3,3],[2,1]] => [2,1] => 2
[3,1,1,1] => [[3,1,1,1],[]] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 2
[2,2,2] => [[2,2,2],[]] => [[2,2,2],[]] => [] => 0
[2,2,1,1] => [[2,2,1,1],[]] => [[2,2,2,2],[1,1]] => [1,1] => 1
[2,1,1,1,1] => [[2,1,1,1,1],[]] => [[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],[]] => [[1,1,1,1,1,1],[]] => [] => 0
[7] => [[7],[]] => [[7],[]] => [] => 0
[6,1] => [[6,1],[]] => [[6,6],[5]] => [5] => 5
[5,2] => [[5,2],[]] => [[5,5],[3]] => [3] => 3
[5,1,1] => [[5,1,1],[]] => [[5,5,5],[4,4]] => [4,4] => 4
[4,3] => [[4,3],[]] => [[4,4],[1]] => [1] => 1
[4,2,1] => [[4,2,1],[]] => [[4,4,4],[3,2]] => [3,2] => 3
[4,1,1,1] => [[4,1,1,1],[]] => [[4,4,4,4],[3,3,3]] => [3,3,3] => 3
[3,3,1] => [[3,3,1],[]] => [[3,3,3],[2]] => [2] => 2
[3,2,2] => [[3,2,2],[]] => [[3,3,3],[1,1]] => [1,1] => 1
[3,2,1,1] => [[3,2,1,1],[]] => [[3,3,3,3],[2,2,1]] => [2,2,1] => 2
[3,1,1,1,1] => [[3,1,1,1,1],[]] => [[3,3,3,3,3],[2,2,2,2]] => [2,2,2,2] => 2
[2,2,2,1] => [[2,2,2,1],[]] => [[2,2,2,2],[1]] => [1] => 1
[2,2,1,1,1] => [[2,2,1,1,1],[]] => [[2,2,2,2,2],[1,1,1]] => [1,1,1] => 1
[2,1,1,1,1,1] => [[2,1,1,1,1,1],[]] => [[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],[]] => [[1,1,1,1,1,1,1],[]] => [] => 0
[8] => [[8],[]] => [[8],[]] => [] => 0
[7,1] => [[7,1],[]] => [[7,7],[6]] => [6] => 6
[6,2] => [[6,2],[]] => [[6,6],[4]] => [4] => 4
[6,1,1] => [[6,1,1],[]] => [[6,6,6],[5,5]] => [5,5] => 5
[5,3] => [[5,3],[]] => [[5,5],[2]] => [2] => 2
[5,2,1] => [[5,2,1],[]] => [[5,5,5],[4,3]] => [4,3] => 4
[5,1,1,1] => [[5,1,1,1],[]] => [[5,5,5,5],[4,4,4]] => [4,4,4] => 4
[4,4] => [[4,4],[]] => [[4,4],[]] => [] => 0
[4,3,1] => [[4,3,1],[]] => [[4,4,4],[3,1]] => [3,1] => 3
[4,2,2] => [[4,2,2],[]] => [[4,4,4],[2,2]] => [2,2] => 2
[4,2,1,1] => [[4,2,1,1],[]] => [[4,4,4,4],[3,3,2]] => [3,3,2] => 3
[4,1,1,1,1] => [[4,1,1,1,1],[]] => [[4,4,4,4,4],[3,3,3,3]] => [3,3,3,3] => 3
[3,3,2] => [[3,3,2],[]] => [[3,3,3],[1]] => [1] => 1
[3,3,1,1] => [[3,3,1,1],[]] => [[3,3,3,3],[2,2]] => [2,2] => 2
[3,2,2,1] => [[3,2,2,1],[]] => [[3,3,3,3],[2,1,1]] => [2,1,1] => 2
[3,2,1,1,1] => [[3,2,1,1,1],[]] => [[3,3,3,3,3],[2,2,2,1]] => [2,2,2,1] => 2
[3,1,1,1,1,1] => [[3,1,1,1,1,1],[]] => [[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],[]] => [] => 0
[2,2,2,1,1] => [[2,2,2,1,1],[]] => [[2,2,2,2,2],[1,1]] => [1,1] => 1
[2,2,1,1,1,1] => [[2,2,1,1,1,1],[]] => [[2,2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 1
[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]] => [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
[9] => [[9],[]] => [[9],[]] => [] => 0
[8,1] => [[8,1],[]] => [[8,8],[7]] => [7] => 7
[7,2] => [[7,2],[]] => [[7,7],[5]] => [5] => 5
[7,1,1] => [[7,1,1],[]] => [[7,7,7],[6,6]] => [6,6] => 6
[6,3] => [[6,3],[]] => [[6,6],[3]] => [3] => 3
[6,2,1] => [[6,2,1],[]] => [[6,6,6],[5,4]] => [5,4] => 5
[6,1,1,1] => [[6,1,1,1],[]] => [[6,6,6,6],[5,5,5]] => [5,5,5] => 5
[5,4] => [[5,4],[]] => [[5,5],[1]] => [1] => 1
[5,3,1] => [[5,3,1],[]] => [[5,5,5],[4,2]] => [4,2] => 4
[5,2,2] => [[5,2,2],[]] => [[5,5,5],[3,3]] => [3,3] => 3
[5,2,1,1] => [[5,2,1,1],[]] => [[5,5,5,5],[4,4,3]] => [4,4,3] => 4
[5,1,1,1,1] => [[5,1,1,1,1],[]] => [[5,5,5,5,5],[4,4,4,4]] => [4,4,4,4] => 4
[4,4,1] => [[4,4,1],[]] => [[4,4,4],[3]] => [3] => 3
[4,3,2] => [[4,3,2],[]] => [[4,4,4],[2,1]] => [2,1] => 2
[4,3,1,1] => [[4,3,1,1],[]] => [[4,4,4,4],[3,3,1]] => [3,3,1] => 3
[4,2,2,1] => [[4,2,2,1],[]] => [[4,4,4,4],[3,2,2]] => [3,2,2] => 3
[4,2,1,1,1] => [[4,2,1,1,1],[]] => [[4,4,4,4,4],[3,3,3,2]] => [3,3,3,2] => 3
[4,1,1,1,1,1] => [[4,1,1,1,1,1],[]] => [[4,4,4,4,4,4],[3,3,3,3,3]] => [3,3,3,3,3] => 3
[3,3,3] => [[3,3,3],[]] => [[3,3,3],[]] => [] => 0
[3,3,2,1] => [[3,3,2,1],[]] => [[3,3,3,3],[2,1]] => [2,1] => 2
[3,3,1,1,1] => [[3,3,1,1,1],[]] => [[3,3,3,3,3],[2,2,2]] => [2,2,2] => 2
[3,2,2,2] => [[3,2,2,2],[]] => [[3,3,3,3],[1,1,1]] => [1,1,1] => 1
[3,2,2,1,1] => [[3,2,2,1,1],[]] => [[3,3,3,3,3],[2,2,1,1]] => [2,2,1,1] => 2
[3,2,1,1,1,1] => [[3,2,1,1,1,1],[]] => [[3,3,3,3,3,3],[2,2,2,2,1]] => [2,2,2,2,1] => 2
[3,1,1,1,1,1,1] => [[3,1,1,1,1,1,1],[]] => [[3,3,3,3,3,3,3],[2,2,2,2,2,2]] => [2,2,2,2,2,2] => 2
[2,2,2,2,1] => [[2,2,2,2,1],[]] => [[2,2,2,2,2],[1]] => [1] => 1
[2,2,2,1,1,1] => [[2,2,2,1,1,1],[]] => [[2,2,2,2,2,2],[1,1,1]] => [1,1,1] => 1
[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]] => [1,1,1,1,1] => 1
[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]] => [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],[]] => [] => 0
[10] => [[10],[]] => [[10],[]] => [] => 0
[9,1] => [[9,1],[]] => [[9,9],[8]] => [8] => 8
[8,2] => [[8,2],[]] => [[8,8],[6]] => [6] => 6
[8,1,1] => [[8,1,1],[]] => [[8,8,8],[7,7]] => [7,7] => 7
[7,3] => [[7,3],[]] => [[7,7],[4]] => [4] => 4
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Description
The largest part of an integer 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.
Map
inner shape
Description
The inner shape of a skew partition.
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