- FindStat

Identifier

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000386: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 8 + 6\ q$

$F_{5} = 16 + 24\ q + 2\ q^{2}$

$F_{6} = 32 + 80\ q + 20\ q^{2}$

$F_{7} = 64 + 240\ q + 120\ q^{2} + 5\ q^{3}$

Description

The number of factors DDU in 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 $r$ .