- FindStat

Identifier

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000459: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The hook length of the base cell of a partition.
This is also known as the perimeter of a partition. In particular, the perimeter of the empty partition is zero.

Map

weight

Description

The weight of a semistandard tableau as an integer partition.
The weight (or content) of a semistandard tableaux $T$ with maximal entry $m$ is the weak composition $(\alpha_1, \dots, \alpha_m)$ such that $\alpha_i$ is the number of letters $i$ occurring in $T$.
This map returns the integer partition obtained by sorting the weight into decreasing order and omitting zeros.
Since semistandard tableaux are bigraded by the size of the partition and the maximal occurring entry, this map is not graded.