- FindStat

Identifier

Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001179: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[2,2,2,2]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 6

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

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

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

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

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

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

[[3,5]] => [1,1] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => 4

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

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

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

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

[[3],[5]] => [1,1] => [1,0,1,1,0,0] => [1,0,1,1,0,0] => 4

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

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

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

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

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

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

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

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

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

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

[[4,4,4]] => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[3,3,3,3]] => [4] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$F_{(2, 3)} = 5\ q^{4}$

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

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

$F_{(3, 3)} = 12\ q^{4} + q^{5}$

$F_{(4, 2)} = 4\ q^{4} + 3\ q^{5} + q^{6}$

$F_{(2, 5)} = 9\ q^{4}$

$F_{(3, 4)} = 24\ q^{4} + q^{5}$

$F_{(4, 3)} = 15\ q^{3} + 8\ q^{4} + 6\ q^{5} + q^{6}$

$F_{(5, 2)} = 10\ q^{5} + q^{7}$

$F_{(2, 6)} = 11\ q^{4}$

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

$F_{(4, 4)} = 45\ q^{3} + 12\ q^{4} + 19\ q^{5} + q^{6}$

$F_{(5, 3)} = 56\ q^{5} + q^{7}$

$F_{(2, 7)} = 13\ q^{4}$

$F_{(3, 6)} = 60\ q^{4} + q^{5}$

$F_{(4, 5)} = 90\ q^{3} + 16\ q^{4} + 52\ q^{5} + q^{6}$

$F_{(5, 4)} = 52\ q^{4} + 138\ q^{5} + q^{7}$

$F_{(2, 8)} = 15\ q^{4}$

$F_{(3, 7)} = 84\ q^{4} + q^{5}$

$F_{(4, 6)} = 150\ q^{3} + 20\ q^{4} + 115\ q^{5} + q^{6}$

$F_{(5, 5)} = 208\ q^{4} + 256\ q^{5} + 26\ q^{6} + q^{7}$

Description

Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra.

Map

to Dyck path

Description

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

Map

swap returns and last descent

Description

Return a Dyck path with number of returns and length of the last descent interchanged.
This is the specialisation of the map

$\Phi$ in [1] to Dyck paths. It is characterised by the fact that the number of up steps before a down step that is neither a return nor part of the last descent is preserved.

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.