Identifier
-
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001292: Dyck paths ⟶ ℤ
Values
[1] => [1,0] => [1,0] => 0
[1,1] => [1,0,1,0] => [1,1,0,0] => 0
[2] => [1,1,0,0] => [1,0,1,0] => 0
[1,1,1] => [1,0,1,0,1,0] => [1,1,0,1,0,0] => 0
[1,2] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 0
[2,1] => [1,1,0,0,1,0] => [1,1,1,0,0,0] => 0
[3] => [1,1,1,0,0,0] => [1,0,1,1,0,0] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => 0
[1,2,1] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 0
[1,3] => [1,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0] => 0
[2,1,1] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0] => 0
[2,2] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 0
[3,1] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,0] => 0
[4] => [1,1,1,1,0,0,0,0] => [1,0,1,1,1,0,0,0] => 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,1,0,0] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,1,0,0] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 0
[5] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,1,0,0] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,1,1,1,0,0,0,0] => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,1,0,0] => 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => 0
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 0
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 0
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,0,1,1,1,1,1,0,0,0,0,0] => 0
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Description
The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Here $A$ is the Nakayama algebra associated to a Dyck path as given in DyckPaths/NakayamaAlgebras.
Here $A$ is the Nakayama algebra associated to a Dyck path as given in DyckPaths/NakayamaAlgebras.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
promotion
Description
The promotion of the two-row standard Young tableau of a Dyck path.
Dyck paths of semilength $n$ are in bijection with standard Young tableaux of shape $(n^2)$, see Mp00033to two-row standard tableau.
This map is the bijection on such standard Young tableaux given by Schützenberger's promotion. For definitions and details, see [1] and the references therein.
Dyck paths of semilength $n$ are in bijection with standard Young tableaux of shape $(n^2)$, see Mp00033to two-row standard tableau.
This map is the bijection on such standard Young tableaux given by Schützenberger's promotion. For definitions and details, see [1] and the references therein.
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