- FindStat

Identifier

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000660: Dyck paths ⟶ ℤ

Values

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

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

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

[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] => 0

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

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

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

[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] => 0

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

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

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

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

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

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

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

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

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

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

[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] => 0

[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] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

[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] => 2

[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] => 0

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

[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] => 0

[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] => 0

[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] => 0

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 9 + 5\ q$

$F_{5} = 21 + 21\ q$

$F_{6} = 51 + 78\ q + 3\ q^{2}$

$F_{7} = 127 + 274\ q + 28\ q^{2}$

Description

The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].

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

$1 \tilde D_2 0 \tilde D_1$ .