- FindStat

Identifier

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000659: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 1 + q$

$F_{3} = 1 + 4\ q$

$F_{4} = 1 + 11\ q + 2\ q^{2}$

$F_{5} = 1 + 26\ q + 15\ q^{2}$

$F_{6} = 1 + 57\ q + 69\ q^{2} + 5\ q^{3}$

$F_{7} = 1 + 120\ q + 252\ q^{2} + 56\ q^{3}$

Description

The number of rises of length at least 2 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 $r$ .