- FindStat

Identifier

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 8 + 6\ q$

$F_{5} = 16 + 24\ q + 2\ q^{2}$

$F_{6} = 32 + 80\ q + 20\ q^{2}$

$F_{7} = 64 + 240\ q + 120\ q^{2} + 5\ q^{3}$

Description

The number of inner corners of the upper path of the parallelogram polyomino associated with the 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 $r$ .