Identifier
-
Mp00230:
Integer partitions
—parallelogram polyomino⟶
Dyck paths
St000011: Dyck paths ⟶ ℤ
Values
[1] => [1,0] => 1
[2] => [1,0,1,0] => 2
[1,1] => [1,1,0,0] => 1
[3] => [1,0,1,0,1,0] => 3
[2,1] => [1,0,1,1,0,0] => 2
[1,1,1] => [1,1,0,1,0,0] => 1
[4] => [1,0,1,0,1,0,1,0] => 4
[3,1] => [1,0,1,0,1,1,0,0] => 3
[2,2] => [1,1,1,0,0,0] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => 2
[1,1,1,1] => [1,1,0,1,0,1,0,0] => 1
[5] => [1,0,1,0,1,0,1,0,1,0] => 5
[4,1] => [1,0,1,0,1,0,1,1,0,0] => 4
[3,2] => [1,0,1,1,1,0,0,0] => 2
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 3
[2,2,1] => [1,1,1,0,0,1,0,0] => 1
[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] => 1
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5
[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] => 4
[3,3] => [1,1,1,0,1,0,0,0] => 1
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 2
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 3
[2,2,2] => [1,1,1,1,0,0,0,0] => 1
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 1
[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] => 1
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 7
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 6
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 5
[4,3] => [1,0,1,1,1,0,1,0,0,0] => 2
[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] => 4
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 1
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 2
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 2
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 3
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 1
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 1
[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] => 1
[8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 8
[7,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 7
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 5
[6,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 6
[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] => 4
[5,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 5
[4,4] => [1,1,1,0,1,0,1,0,0,0] => 1
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 2
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3
[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] => 4
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => 1
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 1
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 2
[3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 2
[3,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 3
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 1
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => 1
[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 1
[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] => 1
[9] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 9
[8,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 8
[7,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 6
[7,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 7
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 4
[6,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 5
[6,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 6
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 2
[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] => 4
[5,1,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 5
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => 1
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => 2
[4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 2
[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => 3
[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] => 4
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => 1
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => 1
[3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => 1
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => 2
[3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => 2
[3,2,1,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 2
[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] => 3
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => 1
[2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 1
[2,2,1,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => 1
[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] => 1
[10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 10
[9,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 9
[8,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 7
[8,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 8
[7,3] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 5
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Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
This is the number of points, excluding the origin, where the Dyck path has height 0.
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.
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.
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