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Identifier

Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000024: Dyck paths ⟶ ℤ (values match St000443The number of long tunnels of a Dyck path., St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path., St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra., St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 4\ q$

$F_{4} = 1 + 11\ q + 3\ q^{2}$

$F_{5} = 1 + 26\ q + 25\ q^{2}$

$F_{6} = 1 + 57\ q + 130\ q^{2} + 15\ q^{3}$

$F_{7} = 1 + 120\ q + 546\ q^{2} + 210\ q^{3}$

Description

The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.

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

shape

Description

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