Identifier
-
Mp00102:
Dyck paths
—rise composition⟶
Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ
Values
[1,0] => [1] => [1] => [1,0] => 0
[1,0,1,0] => [1,1] => [1,1] => [1,0,1,0] => 1
[1,1,0,0] => [2] => [2] => [1,1,0,0] => 0
[1,0,1,1,0,0] => [1,2] => [2,1] => [1,1,0,0,1,0] => 1
[1,1,0,0,1,0] => [2,1] => [1,2] => [1,0,1,1,0,0] => 2
[1,1,0,1,0,0] => [2,1] => [1,2] => [1,0,1,1,0,0] => 2
[1,1,1,0,0,0] => [3] => [3] => [1,1,1,0,0,0] => 0
[1,0,1,1,1,0,0,0] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0] => 1
[1,1,0,0,1,0,1,0] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => 3
[1,1,0,0,1,1,0,0] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[1,1,0,1,0,0,1,0] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => 3
[1,1,0,1,0,1,0,0] => [2,1,1] => [1,2,1] => [1,0,1,1,0,0,1,0] => 3
[1,1,0,1,1,0,0,0] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0] => 2
[1,1,1,0,0,0,1,0] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0] => 3
[1,1,1,0,0,1,0,0] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0] => 3
[1,1,1,0,1,0,0,0] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0] => 3
[1,1,1,1,0,0,0,0] => [4] => [4] => [1,1,1,1,0,0,0,0] => 0
[1,0,1,1,1,1,0,0,0,0] => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 1
[1,1,0,0,1,0,1,1,0,0] => [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
[1,1,0,0,1,1,0,0,1,0] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,0,1,1,0,1,0,0] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,0,1,1,1,0,0,0] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
[1,1,0,1,0,0,1,1,0,0] => [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
[1,1,0,1,0,1,1,0,0,0] => [2,1,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 3
[1,1,0,1,1,0,0,0,1,0] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,1,1,0,0,1,0,0] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,1,1,0,1,0,0,0] => [2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,1,1,1,0,0,0,0] => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 2
[1,1,1,0,0,0,1,0,1,0] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
[1,1,1,0,0,0,1,1,0,0] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3
[1,1,1,0,0,1,0,0,1,0] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
[1,1,1,0,0,1,0,1,0,0] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
[1,1,1,0,0,1,1,0,0,0] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3
[1,1,1,0,1,0,0,0,1,0] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
[1,1,1,0,1,0,0,1,0,0] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
[1,1,1,0,1,0,1,0,0,0] => [3,1,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 4
[1,1,1,0,1,1,0,0,0,0] => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 3
[1,1,1,1,0,0,0,0,1,0] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 4
[1,1,1,1,0,0,0,1,0,0] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 4
[1,1,1,1,0,0,1,0,0,0] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 4
[1,1,1,1,0,1,0,0,0,0] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 4
[1,1,1,1,1,0,0,0,0,0] => [5] => [5] => [1,1,1,1,1,0,0,0,0,0] => 0
[1,0,1,1,1,1,1,0,0,0,0,0] => [1,5] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,1,0,0,1,1,0,0,1,1,0,0] => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,1,0,0,1,1,0,1,1,0,0,0] => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,0,1,1,1,0,0,0,1,0] => [2,3,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,3,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,3,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,1,0,0,1,1,1,1,0,0,0,0] => [2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
[1,1,0,1,0,0,1,1,1,0,0,0] => [2,1,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,1,3] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 3
[1,1,0,1,1,0,0,0,1,0,1,0] => [2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,1,0,1,1,0,0,0,1,1,0,0] => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,1,1,0,0,1,0,0,1,0] => [2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,1,0,1,1,0,0,1,0,1,0,0] => [2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,1,0,1,1,0,0,1,1,0,0,0] => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,1,1,0,1,0,0,0,1,0] => [2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,1,0,1,1,0,1,0,0,1,0,0] => [2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,1,0,1,1,0,1,0,1,0,0,0] => [2,2,1,1] => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 5
[1,1,0,1,1,0,1,1,0,0,0,0] => [2,2,2] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,1,1,1,0,0,0,0,1,0] => [2,3,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,1,0,1,1,1,0,0,0,1,0,0] => [2,3,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,1,0,1,1,1,0,0,1,0,0,0] => [2,3,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,3,1] => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,4] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 2
[1,1,1,0,0,0,1,0,1,1,0,0] => [3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,1,1,0,0,0,1,1,0,1,0,0] => [3,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,1,1,0,0,0,1,1,1,0,0,0] => [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[1,1,1,0,0,1,0,0,1,1,0,0] => [3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[1,1,1,0,0,1,0,1,1,0,0,0] => [3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[1,1,1,0,0,1,1,0,0,0,1,0] => [3,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,1,1,0,0,1,1,0,0,1,0,0] => [3,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[1,1,1,0,1,0,0,0,1,1,0,0] => [3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[1,1,1,0,1,0,1,1,0,0,0,0] => [3,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 4
[1,1,1,0,1,1,0,0,0,0,1,0] => [3,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,1,1,0,1,1,0,0,0,1,0,0] => [3,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,1,1,0,1,1,0,0,1,0,0,0] => [3,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,1,1,0,1,1,0,1,0,0,0,0] => [3,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,1,1,0,1,1,1,0,0,0,0,0] => [3,3] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
[1,1,1,1,0,0,0,0,1,0,1,0] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,1,1,0,0,0,0,1,1,0,0] => [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[1,1,1,1,0,0,0,1,0,0,1,0] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,1,1,0,0,0,1,1,0,0,0] => [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[1,1,1,1,0,0,1,0,0,0,1,0] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,1,1,0,0,1,0,1,0,0,0] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,1,1,0,0,1,1,0,0,0,0] => [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
[1,1,1,1,0,1,0,0,0,0,1,0] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,1,1,0,1,0,0,1,0,0,0] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,1,1,0,1,0,1,0,0,0,0] => [4,1,1] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 5
[1,1,1,1,0,1,1,0,0,0,0,0] => [4,2] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 4
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Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
rise composition
Description
Send a Dyck path to the composition of sizes of its rises.
Map
rotate back to front
Description
The back to front rotation of an integer composition.
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