- FindStat

Identifier

Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤ

Values

[1,0] => [1,0] => 1

[1,0,1,0] => [1,0,1,0] => 2

[1,1,0,0] => [1,1,0,0] => 1

[1,0,1,0,1,0] => [1,0,1,0,1,0] => 3

[1,0,1,1,0,0] => [1,0,1,1,0,0] => 2

[1,1,0,0,1,0] => [1,1,0,0,1,0] => 2

[1,1,0,1,0,0] => [1,0,1,1,0,0] => 2

[1,1,1,0,0,0] => [1,1,1,0,0,0] => 1

[1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0] => 4

[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] => 2

[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] => 2

[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] => 2

[1,1,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0] => 2

[1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0] => 2

[1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,0] => 2

[1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,0] => 2

[1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0] => 1

[1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => 4

[1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => 4

[1,0,1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,1,0,0] => 4

[1,0,1,0,1,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 3

[1,0,1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => 4

[1,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 3

[1,0,1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0,1,0] => 4

[1,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 3

[1,0,1,1,0,1,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 3

[1,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 3

[1,0,1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 3

[1,0,1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 3

[1,0,1,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 2

[1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,0,1,0,1,0] => 4

[1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => 3

[1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3

[1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,1,0,0] => 3

[1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,1,0,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0] => 4

[1,1,0,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 3

[1,1,0,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3

[1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 3

[1,1,0,1,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 3

[1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 3

[1,1,0,1,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 2

[1,1,1,0,0,0,1,0,1,0] => [1,1,1,0,0,0,1,0,1,0] => 3

[1,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2

[1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,0] => 3

[1,1,1,0,0,1,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2

[1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,1,1,0,1,0,0,0,1,0] => [1,0,1,1,1,0,0,0,1,0] => 3

[1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2

[1,1,1,0,1,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,1,1,0,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 2

[1,1,1,1,0,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,0] => 2

[1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,0] => 2

[1,1,1,1,0,0,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0] => 2

[1,1,1,1,0,1,0,0,0,0] => [1,0,1,1,1,1,0,0,0,0] => 2

[1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => 1

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,0,1,0,1,0,1,0,1,0,1,0] => 6

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0,1,0] => 5

[1,0,1,0,1,0,1,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,1,0,0] => 5

[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] => 5

[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] => 5

[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] => 3

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 5

[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] => 3

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,0,0,1,0,1,0] => 5

[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] => 3

[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] => 3

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3

[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] => 3

[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] => 3

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3

[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] => 3

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3

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Description

The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 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.