Identifier
-
Mp00020:
Binary trees
—to Tamari-corresponding Dyck path⟶
Dyck paths
St000005: Dyck paths ⟶ ℤ
Values
=>
Cc0010;cc-rep-0
Cc0005;cc-rep
[.,.]=>[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
[.,[[[.,[.,[.,.]]],.],.]]=>[1,1,1,1,0,0,0,1,0,1,0,0]=>4
[.,[[[.,[[.,.],.]],.],.]]=>[1,1,1,0,1,0,0,1,0,1,0,0]=>5
[.,[[[[.,.],[.,.]],.],.]]=>[1,1,0,1,1,0,0,1,0,1,0,0]=>5
[.,[[[[.,[.,.]],.],.],.]]=>[1,1,1,0,0,1,0,1,0,1,0,0]=>6
[.,[[[[[.,.],.],.],.],.]]=>[1,1,0,1,0,1,0,1,0,1,0,0]=>6
[[.,.],[.,[.,[.,[.,.]]]]]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>1
[[.,.],[.,[.,[[.,.],.]]]]=>[1,0,1,1,1,1,0,1,0,0,0,0]=>3
[[.,.],[.,[[.,.],[.,.]]]]=>[1,0,1,1,1,0,1,1,0,0,0,0]=>3
[[.,.],[.,[[.,[.,.]],.]]]=>[1,0,1,1,1,1,0,0,1,0,0,0]=>4
[[.,.],[.,[[[.,.],.],.]]]=>[1,0,1,1,1,0,1,0,1,0,0,0]=>4
[[.,.],[[.,.],[.,[.,.]]]]=>[1,0,1,1,0,1,1,1,0,0,0,0]=>3
[[.,.],[[.,.],[[.,.],.]]]=>[1,0,1,1,0,1,1,0,1,0,0,0]=>4
[[.,.],[[.,[.,.]],[.,.]]]=>[1,0,1,1,1,0,0,1,1,0,0,0]=>4
[[.,.],[[[.,.],.],[.,.]]]=>[1,0,1,1,0,1,0,1,1,0,0,0]=>4
[[.,.],[[.,[.,[.,.]]],.]]=>[1,0,1,1,1,1,0,0,0,1,0,0]=>5
[[.,.],[[.,[[.,.],.]],.]]=>[1,0,1,1,1,0,1,0,0,1,0,0]=>5
[[.,.],[[[.,.],[.,.]],.]]=>[1,0,1,1,0,1,1,0,0,1,0,0]=>7
[[.,.],[[[.,[.,.]],.],.]]=>[1,0,1,1,1,0,0,1,0,1,0,0]=>5
[[.,.],[[[[.,.],.],.],.]]=>[1,0,1,1,0,1,0,1,0,1,0,0]=>7
[[.,[.,.]],[.,[.,[.,.]]]]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>2
[[.,[.,.]],[.,[[.,.],.]]]=>[1,1,0,0,1,1,1,0,1,0,0,0]=>5
[[.,[.,.]],[[.,.],[.,.]]]=>[1,1,0,0,1,1,0,1,1,0,0,0]=>5
[[.,[.,.]],[[.,[.,.]],.]]=>[1,1,0,0,1,1,1,0,0,1,0,0]=>6
[[.,[.,.]],[[[.,.],.],.]]=>[1,1,0,0,1,1,0,1,0,1,0,0]=>6
[[[.,.],.],[.,[.,[.,.]]]]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>3
[[[.,.],.],[.,[[.,.],.]]]=>[1,0,1,0,1,1,1,0,1,0,0,0]=>6
[[[.,.],.],[[.,.],[.,.]]]=>[1,0,1,0,1,1,0,1,1,0,0,0]=>6
[[[.,.],.],[[.,[.,.]],.]]=>[1,0,1,0,1,1,1,0,0,1,0,0]=>7
[[[.,.],.],[[[.,.],.],.]]=>[1,0,1,0,1,1,0,1,0,1,0,0]=>7
[[.,[.,[.,.]]],[.,[.,.]]]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>3
[[.,[.,[.,.]]],[[.,.],.]]=>[1,1,1,0,0,0,1,1,0,1,0,0]=>7
[[.,[[.,.],.]],[.,[.,.]]]=>[1,1,0,1,0,0,1,1,1,0,0,0]=>4
[[.,[[.,.],.]],[[.,.],.]]=>[1,1,0,1,0,0,1,1,0,1,0,0]=>8
[[[.,.],[.,.]],[.,[.,.]]]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>4
[[[.,.],[.,.]],[[.,.],.]]=>[1,0,1,1,0,0,1,1,0,1,0,0]=>8
[[[.,[.,.]],.],[.,[.,.]]]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>5
[[[.,[.,.]],.],[[.,.],.]]=>[1,1,0,0,1,0,1,1,0,1,0,0]=>9
[[[[.,.],.],.],[.,[.,.]]]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>6
[[[[.,.],.],.],[[.,.],.]]=>[1,0,1,0,1,0,1,1,0,1,0,0]=>10
[[.,[.,[.,[.,.]]]],[.,.]]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>4
[[.,[.,[[.,.],.]]],[.,.]]=>[1,1,1,0,1,0,0,0,1,1,0,0]=>5
[[.,[[.,.],[.,.]]],[.,.]]=>[1,1,0,1,1,0,0,0,1,1,0,0]=>5
[[.,[[.,[.,.]],.]],[.,.]]=>[1,1,1,0,0,1,0,0,1,1,0,0]=>6
[[.,[[[.,.],.],.]],[.,.]]=>[1,1,0,1,0,1,0,0,1,1,0,0]=>6
[[[.,.],[.,[.,.]]],[.,.]]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>5
[[[.,.],[[.,.],.]],[.,.]]=>[1,0,1,1,0,1,0,0,1,1,0,0]=>7
[[[.,[.,.]],[.,.]],[.,.]]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>6
[[[[.,.],.],[.,.]],[.,.]]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>7
[[[.,[.,[.,.]]],.],[.,.]]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>7
[[[.,[[.,.],.]],.],[.,.]]=>[1,1,0,1,0,0,1,0,1,1,0,0]=>8
[[[[.,.],[.,.]],.],[.,.]]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>8
[[[[.,[.,.]],.],.],[.,.]]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>9
[[[[[.,.],.],.],.],[.,.]]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>10
[[.,[.,[.,[.,[.,.]]]]],.]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>5
[[.,[.,[.,[[.,.],.]]]],.]=>[1,1,1,1,0,1,0,0,0,0,1,0]=>6
[[.,[.,[[.,.],[.,.]]]],.]=>[1,1,1,0,1,1,0,0,0,0,1,0]=>6
[[.,[.,[[.,[.,.]],.]]],.]=>[1,1,1,1,0,0,1,0,0,0,1,0]=>7
[[.,[.,[[[.,.],.],.]]],.]=>[1,1,1,0,1,0,1,0,0,0,1,0]=>7
[[.,[[.,.],[.,[.,.]]]],.]=>[1,1,0,1,1,1,0,0,0,0,1,0]=>6
[[.,[[.,.],[[.,.],.]]],.]=>[1,1,0,1,1,0,1,0,0,0,1,0]=>7
[[.,[[.,[.,.]],[.,.]]],.]=>[1,1,1,0,0,1,1,0,0,0,1,0]=>7
[[.,[[[.,.],.],[.,.]]],.]=>[1,1,0,1,0,1,1,0,0,0,1,0]=>7
[[.,[[.,[.,[.,.]]],.]],.]=>[1,1,1,1,0,0,0,1,0,0,1,0]=>8
[[.,[[.,[[.,.],.]],.]],.]=>[1,1,1,0,1,0,0,1,0,0,1,0]=>8
[[.,[[[.,.],[.,.]],.]],.]=>[1,1,0,1,1,0,0,1,0,0,1,0]=>9
[[.,[[[.,[.,.]],.],.]],.]=>[1,1,1,0,0,1,0,1,0,0,1,0]=>8
[[.,[[[[.,.],.],.],.]],.]=>[1,1,0,1,0,1,0,1,0,0,1,0]=>9
[[[.,.],[.,[.,[.,.]]]],.]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>6
[[[.,.],[.,[[.,.],.]]],.]=>[1,0,1,1,1,0,1,0,0,0,1,0]=>8
[[[.,.],[[.,.],[.,.]]],.]=>[1,0,1,1,0,1,1,0,0,0,1,0]=>8
[[[.,.],[[.,[.,.]],.]],.]=>[1,0,1,1,1,0,0,1,0,0,1,0]=>9
[[[.,.],[[[.,.],.],.]],.]=>[1,0,1,1,0,1,0,1,0,0,1,0]=>9
[[[.,[.,.]],[.,[.,.]]],.]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>7
[[[.,[.,.]],[[.,.],.]],.]=>[1,1,0,0,1,1,0,1,0,0,1,0]=>10
[[[[.,.],.],[.,[.,.]]],.]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>8
[[[[.,.],.],[[.,.],.]],.]=>[1,0,1,0,1,1,0,1,0,0,1,0]=>11
[[[.,[.,[.,.]]],[.,.]],.]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>8
[[[.,[[.,.],.]],[.,.]],.]=>[1,1,0,1,0,0,1,1,0,0,1,0]=>9
[[[[.,.],[.,.]],[.,.]],.]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>9
[[[[.,[.,.]],.],[.,.]],.]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>10
[[[[[.,.],.],.],[.,.]],.]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>11
[[[.,[.,[.,[.,.]]]],.],.]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>9
[[[.,[.,[[.,.],.]]],.],.]=>[1,1,1,0,1,0,0,0,1,0,1,0]=>10
[[[.,[[.,.],[.,.]]],.],.]=>[1,1,0,1,1,0,0,0,1,0,1,0]=>10
[[[.,[[.,[.,.]],.]],.],.]=>[1,1,1,0,0,1,0,0,1,0,1,0]=>11
[[[.,[[[.,.],.],.]],.],.]=>[1,1,0,1,0,1,0,0,1,0,1,0]=>11
[[[[.,.],[.,[.,.]]],.],.]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>10
[[[[.,.],[[.,.],.]],.],.]=>[1,0,1,1,0,1,0,0,1,0,1,0]=>12
[[[[.,[.,.]],[.,.]],.],.]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>11
[[[[[.,.],.],[.,.]],.],.]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>12
[[[[.,[.,[.,.]]],.],.],.]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>12
[[[[.,[[.,.],.]],.],.],.]=>[1,1,0,1,0,0,1,0,1,0,1,0]=>13
[[[[[.,.],[.,.]],.],.],.]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>13
[[[[[.,[.,.]],.],.],.],.]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>14
[[[[[[.,.],.],.],.],.],.]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>15
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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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