- FindStat

Identifier

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000005: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + q^{2} + q^{3}$

$F_{4} = 1 + 3\ q + 3\ q^{2} + 3\ q^{3} + 2\ q^{4} + q^{5} + q^{6}$

$F_{5} = 1 + 4\ q + 6\ q^{2} + 7\ q^{3} + 7\ q^{4} + 5\ q^{5} + 5\ q^{6} + 3\ q^{7} + 2\ q^{8} + q^{9} + q^{10}$

$F_{6} = 1 + 5\ q + 10\ q^{2} + 14\ q^{3} + 17\ q^{4} + 16\ q^{5} + 16\ q^{6} + 14\ q^{7} + 11\ q^{8} + 9\ q^{9} + 7\ q^{10} + 5\ q^{11} + 3\ q^{12} + 2\ q^{13} + q^{14} + q^{15}$

Description

The bounce statistic of a Dyck path.
The bounce path

$D'$ of a Dyck path

$D$ is the Dyck path obtained from

$D$ by starting at the end point

$(2n,0)$ , traveling north-west until hitting

$D$ , then bouncing back south-west to the

$x$ -axis, and repeating this procedure until finally reaching the point

$(0,0)$ .
The points where

$D'$ touches the

$x$ -axis are called bounce points, and a bounce path is uniquely determined by its bounce points.
This statistic is given by the sum of all

$i$ for which the bounce path

$D'$ of

$D$ touches the

$x$ -axis at

$(2i,0)$ .
In particular, the bounce statistics of

$D$ and

$D'$ coincide.

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 $r$ .