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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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.
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:
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 $r$.
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