- FindStat

Identifier

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤ (values match St000442The maximal area to the right of an up step of a Dyck path.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[[[.,.],[.,.]],[.,.]]] => [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] => 3

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

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

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

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

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

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

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

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

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

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

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Description

The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.

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 $r$.