Identifier
-
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
St000439: Dyck paths ⟶ ℤ (values match St000025The number of initial rises of a Dyck path.)
Values
[1] => [1,0] => 2
[1,1] => [1,0,1,0] => 2
[2] => [1,1,0,0] => 3
[1,1,1] => [1,0,1,0,1,0] => 2
[1,2] => [1,0,1,1,0,0] => 2
[2,1] => [1,1,0,0,1,0] => 3
[3] => [1,1,1,0,0,0] => 4
[1,1,1,1] => [1,0,1,0,1,0,1,0] => 2
[1,1,2] => [1,0,1,0,1,1,0,0] => 2
[1,2,1] => [1,0,1,1,0,0,1,0] => 2
[1,3] => [1,0,1,1,1,0,0,0] => 2
[2,1,1] => [1,1,0,0,1,0,1,0] => 3
[2,2] => [1,1,0,0,1,1,0,0] => 3
[3,1] => [1,1,1,0,0,0,1,0] => 4
[4] => [1,1,1,1,0,0,0,0] => 5
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 2
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0] => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 3
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 3
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
[2,3] => [1,1,0,0,1,1,1,0,0,0] => 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 4
[3,2] => [1,1,1,0,0,0,1,1,0,0] => 4
[4,1] => [1,1,1,1,0,0,0,0,1,0] => 5
[5] => [1,1,1,1,1,0,0,0,0,0] => 6
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 2
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 2
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 3
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 3
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 3
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 3
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 3
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 3
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 3
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 3
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 4
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 4
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 4
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 4
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 5
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 5
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 6
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 7
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 2
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 2
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 2
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 2
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 2
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => 2
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 2
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 2
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 2
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => 2
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => 2
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => 2
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 2
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => 2
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 2
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => 2
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 2
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => 2
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => 2
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => 2
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 2
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 2
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 2
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 2
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 2
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 2
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 2
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 2
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 2
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 2
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 2
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 3
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 3
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 3
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => 3
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 3
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => 3
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Description
The position of the first down step of a Dyck path.
Map
bounce path
Description
The bounce path determined by an integer composition.
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