- FindStat

Identifier

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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 Tamari-corresponding Dyck path

Description

Return the Dyck path associated with a binary tree in consistency with the Tamari order on Dyck words and binary trees.
The bijection is defined recursively as follows:

a leaf is associated with an empty Dyck path,
a tree with children $l,r$ is associated with the Dyck word $T(l) 1 T(r) 0$ where $T(l)$ and $T(r)$ are the images of this bijection to $l$ and $r$ .