- FindStat

Identifier

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001526: Dyck paths ⟶ ℤ

Values

{{1}} => [1] => [1,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,5}} => [5] => [1,0,1,0,1,0,1,0,1,0] => 2

{{1,2,3,4},{5}} => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 2

{{1,2,3,5},{4}} => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 2

{{1,2,3},{4,5}} => [3,2] => [1,0,1,1,1,0,0,0] => 3

{{1,2,3},{4},{5}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3

{{1,2,4,5},{3}} => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 2

{{1,2,4},{3,5}} => [3,2] => [1,0,1,1,1,0,0,0] => 3

{{1,2,4},{3},{5}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3

{{1,2,5},{3,4}} => [3,2] => [1,0,1,1,1,0,0,0] => 3

{{1,2},{3,4,5}} => [3,2] => [1,0,1,1,1,0,0,0] => 3

{{1,2},{3,4},{5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 3

{{1,2,5},{3},{4}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3

{{1,2},{3,5},{4}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 3

{{1,2},{3},{4,5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 3

{{1,2},{3},{4},{5}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 3

{{1,3,4,5},{2}} => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 2

{{1,3,4},{2,5}} => [3,2] => [1,0,1,1,1,0,0,0] => 3

{{1,3,4},{2},{5}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3

{{1,3,5},{2,4}} => [3,2] => [1,0,1,1,1,0,0,0] => 3

{{1,3},{2,4,5}} => [3,2] => [1,0,1,1,1,0,0,0] => 3

{{1,3},{2,4},{5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 3

{{1,3,5},{2},{4}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3

{{1,3},{2,5},{4}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 3

{{1,3},{2},{4,5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 3

{{1,3},{2},{4},{5}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 3

{{1,4,5},{2,3}} => [3,2] => [1,0,1,1,1,0,0,0] => 3

{{1,4},{2,3,5}} => [3,2] => [1,0,1,1,1,0,0,0] => 3

{{1,4},{2,3},{5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 3

{{1,5},{2,3,4}} => [3,2] => [1,0,1,1,1,0,0,0] => 3

{{1},{2,3,4,5}} => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 2

{{1},{2,3,4},{5}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3

{{1,5},{2,3},{4}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 3

{{1},{2,3,5},{4}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3

{{1},{2,3},{4,5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 3

{{1},{2,3},{4},{5}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 3

{{1,4,5},{2},{3}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3

{{1,4},{2,5},{3}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 3

{{1,4},{2},{3,5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 3

{{1,4},{2},{3},{5}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 3

{{1,5},{2,4},{3}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 3

{{1},{2,4,5},{3}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3

{{1},{2,4},{3,5}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 3

{{1},{2,4},{3},{5}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 3

{{1,5},{2},{3,4}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 3

{{1},{2,5},{3,4}} => [2,2,1] => [1,1,1,0,0,1,0,0] => 3

{{1},{2},{3,4,5}} => [3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3

{{1},{2},{3,4},{5}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 3

{{1,5},{2},{3},{4}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 3

{{1},{2,5},{3},{4}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 3

{{1},{2},{3,5},{4}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 3

{{1},{2},{3},{4,5}} => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 3

{{1},{2},{3},{4},{5}} => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = 2\ q^{2}$

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

$F_{4} = 5\ q^{2} + 10\ q^{3}$

$F_{5} = 6\ q^{2} + 46\ q^{3}$

Description

The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.

Map

shape

Description

Sends a set partition to the integer partition obtained by the sizes of the blocks.

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.