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Identifier

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
St000185: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[2,5]] => [2] => 0

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

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

[[5,5]] => [2] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

shape

Description

Return the shape of a tableau.