Identifier
-
Mp00179:
Integer partitions
—to skew partition⟶
Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001432: Integer partitions ⟶ ℤ (values match St000783The side length of the largest staircase partition fitting into a partition.)
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
[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
[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
[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
[6,2] => [[6,2],[]] => [[6,6],[4]] => [6,6] => 2
[5,3] => [[5,3],[]] => [[5,5],[2]] => [5,5] => 2
[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
[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
[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
[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
[6,3] => [[6,3],[]] => [[6,6],[3]] => [6,6] => 2
[5,4] => [[5,4],[]] => [[5,5],[1]] => [5,5] => 2
[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
[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
[3,2,2,2] => [[3,2,2,2],[]] => [[3,3,3,3],[1,1,1]] => [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
[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
[6,4] => [[6,4],[]] => [[6,6],[2]] => [6,6] => 2
[5,5] => [[5,5],[]] => [[5,5],[]] => [5,5] => 2
[4,4,2] => [[4,4,2],[]] => [[4,4,4],[2]] => [4,4,4] => 3
[4,3,3] => [[4,3,3],[]] => [[4,4,4],[1,1]] => [4,4,4] => 3
[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
[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
[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
[6,5] => [[6,5],[]] => [[6,6],[1]] => [6,6] => 2
[4,4,3] => [[4,4,3],[]] => [[4,4,4],[1]] => [4,4,4] => 3
[3,3,3,2] => [[3,3,3,2],[]] => [[3,3,3,3],[1]] => [3,3,3,3] => 3
[6,6] => [[6,6],[]] => [[6,6],[]] => [6,6] => 2
[4,4,4] => [[4,4,4],[]] => [[4,4,4],[]] => [4,4,4] => 3
[3,3,3,3] => [[3,3,3,3],[]] => [[3,3,3,3],[]] => [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
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Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
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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