- FindStat

Identifier

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000659: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

[.,[.,[.,[.,.]]]] => [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] => 2

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,.]]]]] => [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] => 2

[.,[.,[[.,[.,.]],.]]] => [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] => 2

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

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

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

[.,[[.,[.,[.,.]]],.]] => [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] => 2

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

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

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

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

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

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

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

[[.,[.,[.,.]]],[.,.]] => [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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,[.,.]]]]]] => [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] => 2

[.,[.,[.,[[.,[.,.]],.]]]] => [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] => 2

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

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

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

[.,[.,[[.,[.,[.,.]]],.]]] => [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] => 2

[.,[.,[[[.,[.,.]],.],.]]] => [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] => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[[.,[.,[.,[.,.]]]],.]] => [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] => 2

[.,[[.,[[.,[.,.]],.]],.]] => [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] => 2

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

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

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

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

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$F_{2} = 1 + q$

$F_{3} = 1 + 4\ q$

$F_{4} = 1 + 11\ q + 2\ q^{2}$

$F_{5} = 1 + 26\ q + 15\ q^{2}$

$F_{6} = 1 + 57\ q + 69\ q^{2} + 5\ q^{3}$

$F_{7} = 1 + 120\ q + 252\ q^{2} + 56\ q^{3}$

Description

The number of rises of length at least 2 of a Dyck path.

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 $r$ .