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Identifier

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000013: Dyck paths ⟶ ℤ (values match St000442The maximal area to the right of an up step of a Dyck path.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 1

[8,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2

[7,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 1

[9,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2

[8,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 3

[8,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 2

[7,3] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 3

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Description

The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.

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.