Identifier
-
Mp00099:
Dyck paths
—bounce path⟶
Dyck paths
St001226: Dyck paths ⟶ ℤ
Values
[1,0] => [1,0] => 2
[1,0,1,0] => [1,0,1,0] => 2
[1,1,0,0] => [1,1,0,0] => 3
[1,0,1,0,1,0] => [1,0,1,0,1,0] => 2
[1,0,1,1,0,0] => [1,0,1,1,0,0] => 3
[1,1,0,0,1,0] => [1,1,0,0,1,0] => 3
[1,1,0,1,0,0] => [1,0,1,1,0,0] => 3
[1,1,1,0,0,0] => [1,1,1,0,0,0] => 4
[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => 2
[1,0,1,0,1,1,0,0] => [1,0,1,0,1,1,0,0] => 3
[1,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 3
[1,0,1,1,0,1,0,0] => [1,0,1,0,1,1,0,0] => 3
[1,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => 4
[1,1,0,0,1,0,1,0] => [1,1,0,0,1,0,1,0] => 3
[1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0] => 4
[1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0] => 3
[1,1,0,1,0,1,0,0] => [1,1,0,0,1,1,0,0] => 4
[1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => 4
[1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 4
[1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => 4
[1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => 4
[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 5
[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => 2
[1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => 3
[1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => 3
[1,0,1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => 3
[1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 4
[1,0,1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => 3
[1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => 3
[1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 4
[1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 4
[1,0,1,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 5
[1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 3
[1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => 4
[1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 4
[1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => 4
[1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 5
[1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => 3
[1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 4
[1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,1,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 5
[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 5
[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 5
[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => 4
[1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 5
[1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 4
[1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 5
[1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 5
[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 4
[1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 5
[1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 5
[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 5
[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 5
[1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 5
[1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 5
[1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 5
[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 6
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 2
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 3
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 3
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 3
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 3
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 3
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 4
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,0,1,1,1,0,0,0] => 4
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 5
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 3
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 4
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 4
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 4
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 3
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 4
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 4
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 5
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0,1,0,1,0] => 4
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 4
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,0,1,0,1,1,1,0,0,0,1,0] => 4
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 5
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 5
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Description
The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra.
That is the number of i such that $Ext_A^1(J,e_i J)=0$.
That is the number of i such that $Ext_A^1(J,e_i J)=0$.
Map
bounce path
Description
Sends a Dyck path $D$ of length $2n$ to its bounce path.
This path is formed by starting at the endpoint $(n,n)$ of $D$ and travelling west until encountering the first vertical step of $D$, then south until hitting the diagonal, then west again to hit $D$, etc. until the point $(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.
This path is formed by starting at the endpoint $(n,n)$ of $D$ and travelling west until encountering the first vertical step of $D$, then south until hitting the diagonal, then west again to hit $D$, etc. until the point $(0,0)$ is reached.
This map is the first part of the zeta map Mp00030zeta map.
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