Identifier
Values
[2] => [1,0,1,0] => 2
[1,1] => [1,1,0,0] => 0
[3] => [1,0,1,0,1,0] => 3
[2,1] => [1,0,1,1,0,0] => 1
[1,1,1] => [1,1,0,1,0,0] => 0
[4] => [1,0,1,0,1,0,1,0] => 4
[3,1] => [1,0,1,0,1,1,0,0] => 2
[2,2] => [1,1,1,0,0,0] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => 0
[5] => [1,0,1,0,1,0,1,0,1,0] => 5
[4,1] => [1,0,1,0,1,0,1,1,0,0] => 3
[3,2] => [1,0,1,1,1,0,0,0] => 1
[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] => 0
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
[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] => 4
[4,2] => [1,0,1,0,1,1,1,0,0,0] => 2
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 3
[3,3] => [1,1,1,0,1,0,0,0] => 0
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1
[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] => 0
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => 0
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 1
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 0
[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] => 5
[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] => 4
[4,3] => [1,0,1,1,1,0,1,0,0,0] => 1
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 2
[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => 3
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => 0
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => 1
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1
[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] => 0
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 0
[2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 1
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 0
[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] => 6
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 4
[6,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => 5
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => 2
[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] => 4
[4,4] => [1,1,1,0,1,0,1,0,0,0] => 0
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 1
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
[4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => 2
[4,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => 3
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => 0
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => 0
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 1
[3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 1
[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] => 0
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => 0
[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 0
[2,1,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 1
[7,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 5
[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] => 4
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 1
[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => 2
[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 3
[5,2,1,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => 3
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => 0
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => 1
[4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 1
[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => 2
[4,2,1,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 2
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => 0
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => 0
[3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => 0
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => 1
[3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => 1
[3,2,1,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => 1
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => 0
[2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 0
[7,3] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 4
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => 2
[6,3,1] => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0] => 3
[6,2,2] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 4
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => 0
[5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => 1
[5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => 2
[5,3,1,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0] => 2
[5,2,2,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0] => 3
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => 0
[4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => 0
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => 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] => [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0] => 1
[4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 2
[4,2,2,1,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0] => 2
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => 0
>>> Load all 190 entries. <<<
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => 0
[3,3,2,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => 0
[3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => 1
[3,2,2,1,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 1
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => 0
[2,2,2,2,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => 0
[7,4] => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0] => 3
[6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => 1
[6,4,1] => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0] => 2
[6,3,2] => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0] => 3
[5,5,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0] => 0
[5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => 1
[5,4,1,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0] => 1
[5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 2
[5,3,2,1] => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 2
[5,2,2,2] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0] => 3
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => 0
[4,4,2,1] => [1,1,1,0,1,0,1,1,0,0,0,1,0,0] => 0
[4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 1
[4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => 1
[4,3,2,1,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0] => 1
[4,2,2,2,1] => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0] => 2
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => 0
[3,3,3,1,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => 0
[3,3,2,2,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0] => 0
[3,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 1
[3,2,2,2,1,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0] => 1
[2,2,2,2,2,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0] => 0
[7,5] => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0] => 2
[6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => 0
[6,5,1] => [1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0] => 1
[6,4,2] => [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0] => 2
[6,3,3] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 3
[5,5,2] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 0
[5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => 1
[5,4,2,1] => [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,0] => 1
[5,3,3,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0] => 2
[5,3,2,2] => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0] => 2
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => 0
[4,4,3,1] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0] => 0
[4,4,2,2] => [1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 0
[4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 1
[4,3,3,1,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0] => 1
[4,3,2,2,1] => [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0] => 1
[4,2,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0] => 2
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
[3,3,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0] => 0
[3,3,2,2,2] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0] => 0
[3,2,2,2,2,1] => [1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0] => 1
[2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0] => 0
[7,6] => [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0] => 1
[6,5,2] => [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0] => 1
[6,4,3] => [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0] => 2
[5,5,3] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0] => 0
[5,4,4] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => 1
[5,4,3,1] => [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0] => 1
[5,4,2,2] => [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0] => 1
[5,3,3,2] => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0] => 2
[4,4,4,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0] => 0
[4,4,3,2] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => 0
[4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 1
[4,3,3,2,1] => [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0] => 1
[4,3,2,2,2] => [1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0] => 1
[3,3,3,3,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 0
[3,3,3,2,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0] => 0
[3,2,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0] => 1
[6,5,3] => [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0] => 1
[6,4,4] => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0] => 2
[5,5,4] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0] => 0
[5,4,4,1] => [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0] => 1
[5,4,3,2] => [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0] => 1
[5,3,3,3] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 2
[4,4,4,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0] => 0
[4,4,3,3] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 0
[4,3,3,3,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 1
[4,3,3,2,2] => [1,0,1,1,1,1,1,0,0,1,0,1,0,0,0,0] => 1
[3,3,3,3,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 0
[6,5,4] => [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0] => 1
[5,5,5] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => 0
[5,4,4,2] => [1,0,1,1,1,1,1,0,1,0,0,1,0,0,0,0] => 1
[5,4,3,3] => [1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0] => 1
[4,4,4,3] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 0
[4,3,3,3,2] => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 0
[6,5,5] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => 1
[5,4,4,3] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 1
[4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
[4,3,3,3,3] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 1
[5,4,4,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 1
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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.