Identifier
-
Mp00230:
Integer partitions
—parallelogram polyomino⟶
Dyck paths
St001034: Dyck paths ⟶ ℤ
Values
[1] => [1,0] => 1
[2] => [1,0,1,0] => 2
[1,1] => [1,1,0,0] => 2
[3] => [1,0,1,0,1,0] => 3
[2,1] => [1,0,1,1,0,0] => 3
[1,1,1] => [1,1,0,1,0,0] => 3
[4] => [1,0,1,0,1,0,1,0] => 4
[3,1] => [1,0,1,0,1,1,0,0] => 4
[2,2] => [1,1,1,0,0,0] => 4
[2,1,1] => [1,0,1,1,0,1,0,0] => 4
[1,1,1,1] => [1,1,0,1,0,1,0,0] => 4
[5] => [1,0,1,0,1,0,1,0,1,0] => 5
[4,1] => [1,0,1,0,1,0,1,1,0,0] => 5
[3,2] => [1,0,1,1,1,0,0,0] => 5
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => 5
[2,2,1] => [1,1,1,0,0,1,0,0] => 5
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 5
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 5
[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] => 6
[4,2] => [1,0,1,0,1,1,1,0,0,0] => 6
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 6
[3,3] => [1,1,1,0,1,0,0,0] => 6
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 6
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 6
[2,2,2] => [1,1,1,1,0,0,0,0] => 6
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 6
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 6
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 6
[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] => 7
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 7
[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 7
[4,3] => [1,0,1,1,1,0,1,0,0,0] => 7
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 7
[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 7
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 7
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 7
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 7
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 7
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 7
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 7
[2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 7
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 7
[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] => 8
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 8
[6,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 8
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 8
[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 8
[5,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 8
[4,4] => [1,1,1,0,1,0,1,0,0,0] => 8
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 8
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => 8
[4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => 8
[4,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 8
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => 8
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 8
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 8
[3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 8
[3,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 8
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => 8
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => 8
[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 8
[2,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 8
[1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => 8
[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] => 9
[7,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 9
[7,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 9
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 9
[6,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 9
[6,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 9
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 9
[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => 9
[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 9
[5,2,1,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => 9
[5,1,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 9
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => 9
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => 9
[4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 9
[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => 9
[4,2,1,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 9
[4,1,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 9
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => 9
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => 9
[3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => 9
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => 9
[3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => 9
[3,2,1,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 9
[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] => 9
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => 9
[2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 9
[2,2,1,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => 9
[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] => 9
[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] => 9
[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] => 10
[8,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 10
[8,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 10
[7,3] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 10
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Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
The (bivariate) generating function is given in [1].
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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