- FindStat

Identifier

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[[.,.],.]]]] => [1,1,1,1,0,1,0,0,0,0] => 8

[.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => 7

[.,[.,[[.,[.,.]],.]]] => [1,1,1,1,0,0,1,0,0,0] => 7

[.,[.,[[[.,.],.],.]]] => [1,1,1,0,1,0,1,0,0,0] => 9

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

[.,[[.,.],[[.,.],.]]] => [1,1,0,1,1,0,1,0,0,0] => 8

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

[.,[[[.,.],.],[.,.]]] => [1,1,0,1,0,1,1,0,0,0] => 7

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

[.,[[.,[[.,.],.]],.]] => [1,1,1,0,1,0,0,1,0,0] => 8

[.,[[[.,.],[.,.]],.]] => [1,1,0,1,1,0,0,1,0,0] => 7

[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => 7

[.,[[[[.,.],.],.],.]] => [1,1,0,1,0,1,0,1,0,0] => 8

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

[[.,.],[.,[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,0] => 7

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

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

[[.,.],[[[.,.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => 7

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

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

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

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

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

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

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

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

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

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

[[.,[.,[[.,.],.]]],.] => [1,1,1,0,1,0,0,0,1,0] => 7

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

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

[[.,[[[.,.],.],.]],.] => [1,1,0,1,0,1,0,0,1,0] => 7

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,[.,.]]]]]] => [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] => 10

[.,[.,[.,[[.,.],[.,.]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => 9

[.,[.,[.,[[.,[.,.]],.]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => 9

[.,[.,[.,[[[.,.],.],.]]]] => [1,1,1,1,0,1,0,1,0,0,0,0] => 12

[.,[.,[[.,.],[.,[.,.]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => 8

[.,[.,[[.,.],[[.,.],.]]]] => [1,1,1,0,1,1,0,1,0,0,0,0] => 11

[.,[.,[[.,[.,.]],[.,.]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => 8

[.,[.,[[[.,.],.],[.,.]]]] => [1,1,1,0,1,0,1,1,0,0,0,0] => 10

[.,[.,[[.,[.,[.,.]]],.]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => 8

[.,[.,[[.,[[.,.],.]],.]]] => [1,1,1,1,0,1,0,0,1,0,0,0] => 11

[.,[.,[[[.,.],[.,.]],.]]] => [1,1,1,0,1,1,0,0,1,0,0,0] => 10

[.,[.,[[[.,[.,.]],.],.]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => 10

[.,[.,[[[[.,.],.],.],.]]] => [1,1,1,0,1,0,1,0,1,0,0,0] => 12

[.,[[.,.],[.,[.,[.,.]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0] => 7

[.,[[.,.],[.,[[.,.],.]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => 10

[.,[[.,.],[[.,.],[.,.]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => 9

[.,[[.,.],[[.,[.,.]],.]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => 9

[.,[[.,.],[[[.,.],.],.]]] => [1,1,0,1,1,0,1,0,1,0,0,0] => 11

[.,[[.,[.,.]],[.,[.,.]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => 7

[.,[[.,[.,.]],[[.,.],.]]] => [1,1,1,0,0,1,1,0,1,0,0,0] => 9

[.,[[[.,.],.],[.,[.,.]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => 8

[.,[[[.,.],.],[[.,.],.]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => 10

[.,[[.,[.,[.,.]]],[.,.]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => 7

[.,[[.,[[.,.],.]],[.,.]]] => [1,1,1,0,1,0,0,1,1,0,0,0] => 9

[.,[[[.,.],[.,.]],[.,.]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => 8

[.,[[[.,[.,.]],.],[.,.]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => 8

[.,[[[[.,.],.],.],[.,.]]] => [1,1,0,1,0,1,0,1,1,0,0,0] => 9

[.,[[.,[.,[.,[.,.]]]],.]] => [1,1,1,1,1,0,0,0,0,1,0,0] => 7

[.,[[.,[.,[[.,.],.]]],.]] => [1,1,1,1,0,1,0,0,0,1,0,0] => 10

[.,[[.,[[.,.],[.,.]]],.]] => [1,1,1,0,1,1,0,0,0,1,0,0] => 9

[.,[[.,[[.,[.,.]],.]],.]] => [1,1,1,1,0,0,1,0,0,1,0,0] => 9

[.,[[.,[[[.,.],.],.]],.]] => [1,1,1,0,1,0,1,0,0,1,0,0] => 11

[.,[[[.,.],[.,[.,.]]],.]] => [1,1,0,1,1,1,0,0,0,1,0,0] => 8

[.,[[[.,.],[[.,.],.]],.]] => [1,1,0,1,1,0,1,0,0,1,0,0] => 10

[.,[[[.,[.,.]],[.,.]],.]] => [1,1,1,0,0,1,1,0,0,1,0,0] => 8

[.,[[[[.,.],.],[.,.]],.]] => [1,1,0,1,0,1,1,0,0,1,0,0] => 9

>>> Load all 1201 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The sum of the heights of the peaks 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 $r$.