- FindStat

Identifier

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000011: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,[.,.]]]]]] => [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] => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1322 entries. <<<

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$F_{1} = q$

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

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

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

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

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

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

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

to Dyck path: up step, left tree, down step, right tree

Description

Return the associated Dyck path, using the bijection 1L0R.
This is given recursively as follows:

a leaf is associated to the empty Dyck Word
a tree with children $l,r$ is associated with the Dyck path described by 1L0R where $L$ and $R$ are respectively the Dyck words associated with the trees $l$ and $r$ .