Identifier
-
Mp00179:
Integer partitions
—to skew partition⟶
Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000783: Integer partitions ⟶ ℤ (values match St001432The order dimension of the 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
[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
[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
[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
[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
[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
[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
[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
[5,5,1] => [[5,5,1],[]] => [[5,5,5],[4]] => [5,5,5] => 3
[5,4,2] => [[5,4,2],[]] => [[5,5,5],[3,1]] => [5,5,5] => 3
[5,3,3] => [[5,3,3],[]] => [[5,5,5],[2,2]] => [5,5,5] => 3
[4,4,3] => [[4,4,3],[]] => [[4,4,4],[1]] => [4,4,4] => 3
[4,4,2,1] => [[4,4,2,1],[]] => [[4,4,4,4],[3,2]] => [4,4,4,4] => 4
[4,3,3,1] => [[4,3,3,1],[]] => [[4,4,4,4],[3,1,1]] => [4,4,4,4] => 4
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Description
The side length of the largest staircase partition fitting into a partition.
For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$.
In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram.
This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$.
In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram.
This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
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
outer shape
Description
The outer shape of the skew partition.
Map
to skew partition
Description
The partition regarded as a skew partition.
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