- FindStat

Identifier

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => [2,3] => 3

[[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => [2,3] => 3

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

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

[[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => [3,2] => 3

[[.,[[.,.],.]],[.,.]] => [1,1,0,1,0,0,1,1,0,0] => [3,2] => 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The largest part of an integer composition.

Map

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.

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 $r$.