- FindStat

Identifier

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.

Map

initial tableau

Description

Sends an integer partition to the standard tableau obtained by filling the numbers $1$ through $n$ row by row.

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.