- FindStat

Identifier

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000335: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$F_{(4, 2)} = 8\ q$

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

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

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

$F_{(5, 2)} = 5\ q + 6\ q^{2}$

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

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

$F_{(4, 4)} = 22\ q + 45\ q^{2} + 10\ q^{4}$

$F_{(5, 3)} = 9\ q + 27\ q^{2} + 21\ q^{3}$

$F_{(6, 2)} = 15\ q$

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

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

$F_{(4, 5)} = 29\ q + 90\ q^{2} + 40\ q^{4}$

$F_{(5, 4)} = 13\ q + 63\ q^{2} + 115\ q^{3}$

$F_{(6, 3)} = 40\ q + 15\ q^{2} + 48\ q^{3}$

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

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

$F_{(4, 6)} = 36\ q + 150\ q^{2} + 100\ q^{4}$

$F_{(5, 5)} = 43\ q + 114\ q^{2} + 334\ q^{3}$

$F_{(6, 4)} = 76\ q + 45\ q^{2} + 200\ q^{3} + 120\ q^{4}$

Description

The difference of lower and upper interactions.
An upper interaction in a Dyck path is the occurrence of a factor

$0^k 1^k$ with

$k \geq 1$ (see St000331The number of upper interactions of a Dyck path.), and a lower interaction is the occurrence of a factor

$1^k 0^k$ with

$k \geq 1$ . In both cases,

$1$ denotes an up-step

$0$ denotes a a down-step.

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

Delest-Viennot

Description

Return the Dyck path corresponding to the parallelogram polyomino obtained by applying Delest-Viennot's bijection.
Let

$D$ be a Dyck path of semilength

$n$ . The parallelogram polyomino

$\gamma(D)$ is defined as follows: let

$\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to

$D$ . Then, the upper path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with even indices, and the lower path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with odd indices.
The Delest-Viennot bijection

$\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path

$(\gamma^{(-1)}\circ\beta)(D)$ .

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.