- FindStat

Identifier

Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤ

Values

[[1]] => [1] => [1,0,1,0] => [1,1,0,1,0,0] => 1
[[1,2]] => [2] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1
[[1],[2]] => [1,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 1
[[1,2,3]] => [3] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 1
[[1,3],[2]] => [2,1] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1
[[1,2],[3]] => [2,1] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 1
[[1],[2],[3]] => [1,1,1] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => 1
[[1,3,4],[2]] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 1
[[1,2,4],[3]] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 1
[[1,2,3],[4]] => [3,1] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => 1
[[1,3],[2,4]] => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 1
[[1,2],[3,4]] => [2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => 1
[[1,4],[2],[3]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 1
[[1,3],[2],[4]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 1
[[1,2],[3],[4]] => [2,1,1] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 1
[[1,3,5],[2,4]] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 1
[[1,2,5],[3,4]] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 1
[[1,3,4],[2,5]] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 1
[[1,2,4],[3,5]] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 1
[[1,2,3],[4,5]] => [3,2] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 1
[[1,4,5],[2],[3]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 1
[[1,3,5],[2],[4]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 1
[[1,2,5],[3],[4]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 1
[[1,3,4],[2],[5]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 1
[[1,2,4],[3],[5]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 1
[[1,2,3],[4],[5]] => [3,1,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 1
[[1,4],[2,5],[3]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[[1,3],[2,5],[4]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[[1,2],[3,5],[4]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[[1,3],[2,4],[5]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[[1,2],[3,4],[5]] => [2,2,1] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => 1
[[1,4,6],[2,5],[3]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[1,3,6],[2,5],[4]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[1,2,6],[3,5],[4]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[1,3,6],[2,4],[5]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[1,2,6],[3,4],[5]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[1,4,5],[2,6],[3]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[1,3,5],[2,6],[4]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[1,2,5],[3,6],[4]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[1,3,4],[2,6],[5]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[1,2,4],[3,6],[5]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[1,2,3],[4,6],[5]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[1,3,5],[2,4],[6]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[1,2,5],[3,4],[6]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[1,3,4],[2,5],[6]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[1,2,4],[3,5],[6]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1
[[1,2,3],[4,5],[6]] => [3,2,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 1

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 4\ q$

Description

The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ .

Map

to Dyck path

Description

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

Map

prime Dyck path

Description

Return the Dyck path obtained by adding an initial up and a final down step.

Map

shape

Description

Sends a tableau to its shape.