- FindStat

Identifier

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{(2, 2)} = 1 + 2\ q$

$F_{(2, 3)} = 1 + 4\ q$

$F_{(3, 2)} = 1 + 4\ q$

$F_{(2, 4)} = 1 + 6\ q$

$F_{(3, 3)} = 1 + 8\ q + 4\ q^{2}$

$F_{(4, 2)} = 1 + 7\ q$

$F_{(2, 5)} = 1 + 8\ q$

$F_{(3, 4)} = 1 + 12\ q + 12\ q^{2}$

$F_{(4, 3)} = 1 + 14\ q + 15\ q^{2}$

$F_{(5, 2)} = 1 + 10\ q$

$F_{(2, 6)} = 1 + 10\ q$

$F_{(3, 5)} = 1 + 16\ q + 24\ q^{2}$

$F_{(4, 4)} = 1 + 21\ q + 45\ q^{2} + 10\ q^{3}$

$F_{(5, 3)} = 1 + 20\ q + 36\ q^{2}$

$F_{(6, 2)} = 1 + 14\ q$

$F_{(2, 7)} = 1 + 12\ q$

$F_{(3, 6)} = 1 + 20\ q + 40\ q^{2}$

$F_{(4, 5)} = 1 + 28\ q + 90\ q^{2} + 40\ q^{3}$

$F_{(5, 4)} = 1 + 30\ q + 108\ q^{2} + 52\ q^{3}$

$F_{(6, 3)} = 1 + 28\ q + 74\ q^{2}$

$F_{(2, 8)} = 1 + 14\ q$

$F_{(3, 7)} = 1 + 24\ q + 60\ q^{2}$

$F_{(4, 6)} = 1 + 35\ q + 150\ q^{2} + 100\ q^{3}$

$F_{(5, 5)} = 1 + 40\ q + 216\ q^{2} + 208\ q^{3} + 26\ q^{4}$

$F_{(6, 4)} = 1 + 42\ q + 222\ q^{2} + 176\ q^{3}$

Description

The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

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.