- FindStat

Identifier

Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000395: 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] => 2

[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] => 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] => 3

[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] => 4

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

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

[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] => 4

[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] => 4

[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] => 4

[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] => 5

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

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

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

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

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

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

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

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

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

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

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

[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] => 5

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

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

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

[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] => 5

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

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

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

[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] => 5

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

[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] => 5

[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] => 5

[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] => 5

[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] => 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] => 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] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1761 entries. <<<

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$F_{1} = q$

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

$F_{3} = 5\ q^{3}$

$F_{4} = 14\ q^{4}$

$F_{5} = 42\ q^{5}$

$F_{6} = 132\ q^{6}$

$F_{7} = 429\ q^{7}$

Description

The sum of the heights of the peaks 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.