- FindStat

Identifier

Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000674: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[7] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,1,1,1] => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 7

[7,1] => [2,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,1,1,1,1,1,1] => [7,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 6

[1,1,1,1,1,1,1,1] => [8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,2,1,1,1,1,1] => [7,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 1 + q^{2}$

$F_{3} = 1 + q + q^{3}$

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

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

$F_{6} = 4 + 2\ q + 2\ q^{2} + q^{3} + q^{4} + q^{6}$

$F_{7} = 4 + 4\ q + 2\ q^{2} + 2\ q^{3} + q^{4} + q^{5} + q^{7}$

Description

The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.

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

conjugate

Description

Return the conjugate partition of the partition.
The conjugate partition of the partition

$\lambda$ of

$n$ is the partition

$\lambda^*$ whose Ferrers diagram is obtained from the diagram of

$\lambda$ by interchanging rows with columns.
This is also called the associated partition or the transpose in the literature.