- FindStat

Identifier

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].

Map

Loehr-Warrington

Description

Return a partition whose diagonal inversion number is the length of the preimage.

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.