- FindStat

Identifier

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000439: Dyck paths ⟶ ℤ (values match St000025The number of initial rises of a Dyck path.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The position of the first down step of a Dyck path.

Map

to Dyck path: up step, left tree, down step, right tree

Description

Return the associated Dyck path, using the bijection 1L0R.
This is given recursively as follows:

a leaf is associated to the empty Dyck Word
a tree with children $l,r$ is associated with the Dyck path described by 1L0R where $L$ and $R$ are respectively the Dyck words associated with the trees $l$ and $r$.