- FindStat

Identifier

Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000225: Integer partitions ⟶ ℤ

Values

[[1],[]] => [[1],[]] => [1] => 0

[[2],[]] => [[2],[]] => [2] => 0

[[1,1],[]] => [[1,1],[]] => [1,1] => 0

[[2,1],[1]] => [[2,1],[1]] => [2,1] => 1

[[3],[]] => [[3],[]] => [3] => 0

[[2,1],[]] => [[2,2],[1]] => [2,2] => 0

[[3,1],[1]] => [[3,2],[2]] => [3,2] => 1

[[2,2],[1]] => [[2,1],[]] => [2,1] => 1

[[3,2],[2]] => [[3,1],[1]] => [3,1] => 2

[[1,1,1],[]] => [[1,1,1],[]] => [1,1,1] => 0

[[2,2,1],[1,1]] => [[2,1,1],[1]] => [2,1,1] => 1

[[2,1,1],[1]] => [[2,2,1],[1,1]] => [2,2,1] => 1

[[3,2,1],[2,1]] => [[3,2,1],[2,1]] => [3,2,1] => 2

[[4],[]] => [[4],[]] => [4] => 0

[[3,1],[]] => [[3,3],[2]] => [3,3] => 0

[[4,1],[1]] => [[4,3],[3]] => [4,3] => 1

[[2,2],[]] => [[2,2],[]] => [2,2] => 0

[[3,2],[1]] => [[3,2],[1]] => [3,2] => 1

[[4,2],[2]] => [[4,2],[2]] => [4,2] => 2

[[2,1,1],[]] => [[2,2,2],[1,1]] => [2,2,2] => 0

[[3,2,1],[1,1]] => [[3,2,2],[2,1]] => [3,2,2] => 1

[[3,1,1],[1]] => [[3,3,2],[2,2]] => [3,3,2] => 1

[[4,2,1],[2,1]] => [[4,3,2],[3,2]] => [4,3,2] => 2

[[3,3],[2]] => [[3,1],[]] => [3,1] => 2

[[4,3],[3]] => [[4,1],[1]] => [4,1] => 3

[[2,2,1],[1]] => [[2,2,1],[1]] => [2,2,1] => 1

[[3,3,1],[2,1]] => [[3,2,1],[2]] => [3,2,1] => 2

[[3,2,1],[2]] => [[3,3,1],[2,1]] => [3,3,1] => 2

[[4,3,1],[3,1]] => [[4,3,1],[3,1]] => [4,3,1] => 3

[[2,2,2],[1,1]] => [[2,1,1],[]] => [2,1,1] => 1

[[3,3,2],[2,2]] => [[3,1,1],[1]] => [3,1,1] => 2

[[3,2,2],[2,1]] => [[3,2,1],[1,1]] => [3,2,1] => 2

[[4,3,2],[3,2]] => [[4,2,1],[2,1]] => [4,2,1] => 3

[[1,1,1,1],[]] => [[1,1,1,1],[]] => [1,1,1,1] => 0

[[2,2,2,1],[1,1,1]] => [[2,1,1,1],[1]] => [2,1,1,1] => 1

[[2,2,1,1],[1,1]] => [[2,2,1,1],[1,1]] => [2,2,1,1] => 1

[[3,3,2,1],[2,2,1]] => [[3,2,1,1],[2,1]] => [3,2,1,1] => 2

[[2,1,1,1],[1]] => [[2,2,2,1],[1,1,1]] => [2,2,2,1] => 1

[[3,2,2,1],[2,1,1]] => [[3,2,2,1],[2,1,1]] => [3,2,2,1] => 2

[[3,2,1,1],[2,1]] => [[3,3,2,1],[2,2,1]] => [3,3,2,1] => 2

[[4,3,2,1],[3,2,1]] => [[4,3,2,1],[3,2,1]] => [4,3,2,1] => 3

[[5],[]] => [[5],[]] => [5] => 0

[[4,1],[]] => [[4,4],[3]] => [4,4] => 0

[[5,1],[1]] => [[5,4],[4]] => [5,4] => 1

[[3,2],[]] => [[3,3],[1]] => [3,3] => 0

[[4,2],[1]] => [[4,3],[2]] => [4,3] => 1

[[5,2],[2]] => [[5,3],[3]] => [5,3] => 2

[[3,1,1],[]] => [[3,3,3],[2,2]] => [3,3,3] => 0

[[4,2,1],[1,1]] => [[4,3,3],[3,2]] => [4,3,3] => 1

[[4,1,1],[1]] => [[4,4,3],[3,3]] => [4,4,3] => 1

[[5,2,1],[2,1]] => [[5,4,3],[4,3]] => [5,4,3] => 2

[[3,3],[1]] => [[3,2],[]] => [3,2] => 1

[[4,3],[2]] => [[4,2],[1]] => [4,2] => 2

[[5,3],[3]] => [[5,2],[2]] => [5,2] => 3

[[2,2,1],[]] => [[2,2,2],[1]] => [2,2,2] => 0

[[3,3,1],[1,1]] => [[3,2,2],[2]] => [3,2,2] => 1

[[3,2,1],[1]] => [[3,3,2],[2,1]] => [3,3,2] => 1

[[4,3,1],[2,1]] => [[4,3,2],[3,1]] => [4,3,2] => 2

[[4,2,1],[2]] => [[4,4,2],[3,2]] => [4,4,2] => 2

[[5,3,1],[3,1]] => [[5,4,2],[4,2]] => [5,4,2] => 3

[[3,2,2],[1,1]] => [[3,2,2],[1,1]] => [3,2,2] => 1

[[4,3,2],[2,2]] => [[4,2,2],[2,1]] => [4,2,2] => 2

[[4,2,2],[2,1]] => [[4,3,2],[2,2]] => [4,3,2] => 2

[[5,3,2],[3,2]] => [[5,3,2],[3,2]] => [5,3,2] => 3

[[2,1,1,1],[]] => [[2,2,2,2],[1,1,1]] => [2,2,2,2] => 0

[[3,2,2,1],[1,1,1]] => [[3,2,2,2],[2,1,1]] => [3,2,2,2] => 1

[[3,2,1,1],[1,1]] => [[3,3,2,2],[2,2,1]] => [3,3,2,2] => 1

[[4,3,2,1],[2,2,1]] => [[4,3,2,2],[3,2,1]] => [4,3,2,2] => 2

[[3,1,1,1],[1]] => [[3,3,3,2],[2,2,2]] => [3,3,3,2] => 1

[[4,2,2,1],[2,1,1]] => [[4,3,3,2],[3,2,2]] => [4,3,3,2] => 2

[[4,2,1,1],[2,1]] => [[4,4,3,2],[3,3,2]] => [4,4,3,2] => 2

[[5,3,2,1],[3,2,1]] => [[5,4,3,2],[4,3,2]] => [5,4,3,2] => 3

[[4,4],[3]] => [[4,1],[]] => [4,1] => 3

[[5,4],[4]] => [[5,1],[1]] => [5,1] => 4

[[3,3,1],[2]] => [[3,3,1],[2]] => [3,3,1] => 2

[[4,4,1],[3,1]] => [[4,3,1],[3]] => [4,3,1] => 3

[[4,3,1],[3]] => [[4,4,1],[3,1]] => [4,4,1] => 3

[[5,4,1],[4,1]] => [[5,4,1],[4,1]] => [5,4,1] => 4

[[2,2,2],[1]] => [[2,2,1],[]] => [2,2,1] => 1

[[3,3,2],[2,1]] => [[3,2,1],[1]] => [3,2,1] => 2

[[4,4,2],[3,2]] => [[4,2,1],[2]] => [4,2,1] => 3

[[3,2,2],[2]] => [[3,3,1],[1,1]] => [3,3,1] => 2

[[4,3,2],[3,1]] => [[4,3,1],[2,1]] => [4,3,1] => 3

[[5,4,2],[4,2]] => [[5,3,1],[3,1]] => [5,3,1] => 4

[[2,2,1,1],[1]] => [[2,2,2,1],[1,1]] => [2,2,2,1] => 1

[[3,3,2,1],[2,1,1]] => [[3,2,2,1],[2,1]] => [3,2,2,1] => 2

[[3,3,1,1],[2,1]] => [[3,3,2,1],[2,2]] => [3,3,2,1] => 2

[[4,4,2,1],[3,2,1]] => [[4,3,2,1],[3,2]] => [4,3,2,1] => 3

[[3,2,1,1],[2]] => [[3,3,3,1],[2,2,1]] => [3,3,3,1] => 2

[[4,3,2,1],[3,1,1]] => [[4,3,3,1],[3,2,1]] => [4,3,3,1] => 3

[[4,3,1,1],[3,1]] => [[4,4,3,1],[3,3,1]] => [4,4,3,1] => 3

[[5,4,2,1],[4,2,1]] => [[5,4,3,1],[4,3,1]] => [5,4,3,1] => 4

[[3,3,3],[2,2]] => [[3,1,1],[]] => [3,1,1] => 2

[[4,4,3],[3,3]] => [[4,1,1],[1]] => [4,1,1] => 3

[[4,3,3],[3,2]] => [[4,2,1],[1,1]] => [4,2,1] => 3

[[5,4,3],[4,3]] => [[5,2,1],[2,1]] => [5,2,1] => 4

[[2,2,2,1],[1,1]] => [[2,2,1,1],[1]] => [2,2,1,1] => 1

[[3,3,3,1],[2,2,1]] => [[3,2,1,1],[2]] => [3,2,1,1] => 2

[[3,3,2,1],[2,2]] => [[3,3,1,1],[2,1]] => [3,3,1,1] => 2

[[4,4,3,1],[3,3,1]] => [[4,3,1,1],[3,1]] => [4,3,1,1] => 3

[[3,2,2,1],[2,1]] => [[3,3,2,1],[2,1,1]] => [3,3,2,1] => 2

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Description

Difference between largest and smallest parts in a 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.

Map

outer shape

Description

The outer shape of the skew partition.