- FindStat

Identifier

Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000185: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

[[1,2,3,4]] => [4] => [] => [] => 0

[[1,3,4],[2]] => [3,1] => [1] => [1] => 0

[[1,2,4],[3]] => [3,1] => [1] => [1] => 0

[[1,2,3],[4]] => [3,1] => [1] => [1] => 0

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

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

[[1,4],[2],[3]] => [2,1,1] => [1,1] => [2] => 0

[[1,3],[2],[4]] => [2,1,1] => [1,1] => [2] => 0

[[1,2],[3],[4]] => [2,1,1] => [1,1] => [2] => 0

[[1],[2],[3],[4]] => [1,1,1,1] => [1,1,1] => [3] => 0

[[1,2,3,4,5]] => [5] => [] => [] => 0

[[1,3,4,5],[2]] => [4,1] => [1] => [1] => 0

[[1,2,4,5],[3]] => [4,1] => [1] => [1] => 0

[[1,2,3,5],[4]] => [4,1] => [1] => [1] => 0

[[1,2,3,4],[5]] => [4,1] => [1] => [1] => 0

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

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

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

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

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

[[1,4,5],[2],[3]] => [3,1,1] => [1,1] => [2] => 0

[[1,3,5],[2],[4]] => [3,1,1] => [1,1] => [2] => 0

[[1,2,5],[3],[4]] => [3,1,1] => [1,1] => [2] => 0

[[1,3,4],[2],[5]] => [3,1,1] => [1,1] => [2] => 0

[[1,2,4],[3],[5]] => [3,1,1] => [1,1] => [2] => 0

[[1,2,3],[4],[5]] => [3,1,1] => [1,1] => [2] => 0

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

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

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

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

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

[[1,5],[2],[3],[4]] => [2,1,1,1] => [1,1,1] => [3] => 0

[[1,4],[2],[3],[5]] => [2,1,1,1] => [1,1,1] => [3] => 0

[[1,3],[2],[4],[5]] => [2,1,1,1] => [1,1,1] => [3] => 0

[[1,2],[3],[4],[5]] => [2,1,1,1] => [1,1,1] => [3] => 0

[[1],[2],[3],[4],[5]] => [1,1,1,1,1] => [1,1,1,1] => [4] => 0

[[1,2,3,4,5,6]] => [6] => [] => [] => 0

[[1,3,4,5,6],[2]] => [5,1] => [1] => [1] => 0

[[1,2,4,5,6],[3]] => [5,1] => [1] => [1] => 0

[[1,2,3,5,6],[4]] => [5,1] => [1] => [1] => 0

[[1,2,3,4,6],[5]] => [5,1] => [1] => [1] => 0

[[1,2,3,4,5],[6]] => [5,1] => [1] => [1] => 0

[[1,3,5,6],[2,4]] => [4,2] => [2] => [1,1] => 1

[[1,2,5,6],[3,4]] => [4,2] => [2] => [1,1] => 1

[[1,3,4,6],[2,5]] => [4,2] => [2] => [1,1] => 1

[[1,2,4,6],[3,5]] => [4,2] => [2] => [1,1] => 1

[[1,2,3,6],[4,5]] => [4,2] => [2] => [1,1] => 1

[[1,3,4,5],[2,6]] => [4,2] => [2] => [1,1] => 1

[[1,2,4,5],[3,6]] => [4,2] => [2] => [1,1] => 1

[[1,2,3,5],[4,6]] => [4,2] => [2] => [1,1] => 1

[[1,2,3,4],[5,6]] => [4,2] => [2] => [1,1] => 1

[[1,4,5,6],[2],[3]] => [4,1,1] => [1,1] => [2] => 0

[[1,3,5,6],[2],[4]] => [4,1,1] => [1,1] => [2] => 0

[[1,2,5,6],[3],[4]] => [4,1,1] => [1,1] => [2] => 0

[[1,3,4,6],[2],[5]] => [4,1,1] => [1,1] => [2] => 0

[[1,2,4,6],[3],[5]] => [4,1,1] => [1,1] => [2] => 0

[[1,2,3,6],[4],[5]] => [4,1,1] => [1,1] => [2] => 0

[[1,3,4,5],[2],[6]] => [4,1,1] => [1,1] => [2] => 0

[[1,2,4,5],[3],[6]] => [4,1,1] => [1,1] => [2] => 0

[[1,2,3,5],[4],[6]] => [4,1,1] => [1,1] => [2] => 0

[[1,2,3,4],[5],[6]] => [4,1,1] => [1,1] => [2] => 0

[[1,3,5],[2,4,6]] => [3,3] => [3] => [1,1,1] => 3

[[1,2,5],[3,4,6]] => [3,3] => [3] => [1,1,1] => 3

[[1,3,4],[2,5,6]] => [3,3] => [3] => [1,1,1] => 3

[[1,2,4],[3,5,6]] => [3,3] => [3] => [1,1,1] => 3

[[1,2,3],[4,5,6]] => [3,3] => [3] => [1,1,1] => 3

[[1,4,6],[2,5],[3]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,3,6],[2,5],[4]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,2,6],[3,5],[4]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,3,6],[2,4],[5]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,2,6],[3,4],[5]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,4,5],[2,6],[3]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,3,5],[2,6],[4]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,2,5],[3,6],[4]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,3,4],[2,6],[5]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,2,4],[3,6],[5]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,2,3],[4,6],[5]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,3,5],[2,4],[6]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,2,5],[3,4],[6]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,3,4],[2,5],[6]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,2,4],[3,5],[6]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,2,3],[4,5],[6]] => [3,2,1] => [2,1] => [2,1] => 1

[[1,5,6],[2],[3],[4]] => [3,1,1,1] => [1,1,1] => [3] => 0

[[1,4,6],[2],[3],[5]] => [3,1,1,1] => [1,1,1] => [3] => 0

[[1,3,6],[2],[4],[5]] => [3,1,1,1] => [1,1,1] => [3] => 0

[[1,2,6],[3],[4],[5]] => [3,1,1,1] => [1,1,1] => [3] => 0

[[1,4,5],[2],[3],[6]] => [3,1,1,1] => [1,1,1] => [3] => 0

[[1,3,5],[2],[4],[6]] => [3,1,1,1] => [1,1,1] => [3] => 0

[[1,2,5],[3],[4],[6]] => [3,1,1,1] => [1,1,1] => [3] => 0

[[1,3,4],[2],[5],[6]] => [3,1,1,1] => [1,1,1] => [3] => 0

[[1,2,4],[3],[5],[6]] => [3,1,1,1] => [1,1,1] => [3] => 0

[[1,2,3],[4],[5],[6]] => [3,1,1,1] => [1,1,1] => [3] => 0

[[1,4],[2,5],[3,6]] => [2,2,2] => [2,2] => [2,2] => 2

[[1,3],[2,5],[4,6]] => [2,2,2] => [2,2] => [2,2] => 2

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Description

The weighted size of a partition.
Let $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ be an integer partition. Then the weighted size of $\lambda$ is
$$\sum_{i=0}^m i \cdot \lambda_i.$$
This is also the sum of the leg lengths of the cells in $\lambda$, or
$$ \sum_i \binom{\lambda^{\prime}_i}{2} $$
where $\lambda^{\prime}$ is the conjugate partition of $\lambda$.
This is the minimal number of inversions a permutation with the given shape can have, see [1, cor.2.2].
This is also the smallest possible sum of the entries of a semistandard tableau (allowing 0 as a part) of shape $\lambda=(\lambda_0,\lambda_1,\ldots,\lambda_m)$, obtained uniquely by placing $i-1$ in all the cells of the $i$th row of $\lambda$, see [2, eq.7.103].

Map

conjugate

Description

Return the conjugate partition of the partition.
The conjugate partition of the partition $\lambda$ of $n$ is the partition $\lambda^*$ whose Ferrers diagram is obtained from the diagram of $\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.

Map

shape

Description

Sends a tableau to its shape.

Map

first row removal

Description

Removes the first entry of an integer partition