- FindStat

Identifier

Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001210: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 3

[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] => 3

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 196 entries. <<<

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$F_{1} = q$

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

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

$F_{4} = 4\ q + 6\ q^{2} + 3\ q^{3} + q^{4}$

$F_{5} = 9\ q + 18\ q^{2} + 10\ q^{3} + 4\ q^{4} + q^{5}$

$F_{6} = 21\ q + 55\ q^{2} + 35\ q^{3} + 15\ q^{4} + 5\ q^{5} + q^{6}$

Description

Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path.

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

$1 \tilde D_2 0 \tilde D_1$ .