- FindStat

Identifier

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[3,5]] => [1,1] => [1,0,1,1,0,0] => 2

[[4,5]] => [1,1] => [1,0,1,1,0,0] => 2

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

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

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

[[3],[5]] => [1,1] => [1,0,1,1,0,0] => 2

[[4],[5]] => [1,1] => [1,0,1,1,0,0] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Description

The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.

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.

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.