- FindStat

Identifier

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1535 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q$

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

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

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

$F_{5} = 14\ q + 14\ q^{2} + 9\ q^{3} + 4\ q^{4} + q^{5}$

$F_{6} = 42\ q + 42\ q^{2} + 28\ q^{3} + 14\ q^{4} + 5\ q^{5} + q^{6}$

$F_{7} = 132\ q + 132\ q^{2} + 90\ q^{3} + 48\ q^{4} + 20\ q^{5} + 6\ q^{6} + q^{7}$

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

decomposition reverse

Description

This map is recursively defined as follows.
The unique empty path of semilength

$0$ is sent to itself.
Let

$D$ be a Dyck path of semilength

$n > 0$ and decompose it into

$1 D_1 0 D_2$ with Dyck paths

$D_1, D_2$ of respective semilengths

$n_1$ and

$n_2$ such that

$n_1$ is minimal. One then has

$n_1+n_2 = n-1$ .
Now let

$\tilde D_1$ and

$\tilde D_2$ be the recursively defined respective images of

$D_1$ and

$D_2$ under this map. The image of

$D$ is then defined as

$1 \tilde D_2 0 \tilde D_1$ .