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Identifier

Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001233: Dyck paths ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 6\ q + q^{3}$

Description

The number of indecomposable 2-dimensional modules with projective dimension one.

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

to semistandard tableau via monotone triangles

Description

The semistandard tableau corresponding the monotone triangle of an alternating sign matrix.
This is obtained by interpreting each row of the monotone triangle as an integer partition, and filling the cells of the smallest partition with ones, the second smallest with twos, and so on.

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.