- FindStat

Identifier

Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

[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] => 1

[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] => 1

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

[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] => 1

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

[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] => 1

[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] => 3

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

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

[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] => 1

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

[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] => 1

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 625 entries. <<<

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = 3\ q + q^{2} + q^{3}$

$F_{4} = 8\ q + 4\ q^{2} + q^{3} + q^{4}$

$F_{5} = 23\ q + 12\ q^{2} + 5\ q^{3} + q^{4} + q^{5}$

$F_{6} = 71\ q + 36\ q^{2} + 17\ q^{3} + 6\ q^{4} + q^{5} + q^{6}$

$F_{7} = 229\ q + 114\ q^{2} + 54\ q^{3} + 23\ q^{4} + 7\ q^{5} + q^{6} + q^{7}$

Description

The position of the first return of a Dyck path.

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.