Identifier
Identifier
Values
[1,0] generating graphics... => 0
[1,0,1,0] generating graphics... => 1
[1,1,0,0] generating graphics... => 0
[1,0,1,0,1,0] generating graphics... => 3
[1,0,1,1,0,0] generating graphics... => 1
[1,1,0,0,1,0] generating graphics... => 2
[1,1,0,1,0,0] generating graphics... => 1
[1,1,1,0,0,0] generating graphics... => 0
[1,0,1,0,1,0,1,0] generating graphics... => 6
[1,0,1,0,1,1,0,0] generating graphics... => 3
[1,0,1,1,0,0,1,0] generating graphics... => 4
[1,0,1,1,0,1,0,0] generating graphics... => 3
[1,0,1,1,1,0,0,0] generating graphics... => 1
[1,1,0,0,1,0,1,0] generating graphics... => 5
[1,1,0,0,1,1,0,0] generating graphics... => 2
[1,1,0,1,0,0,1,0] generating graphics... => 4
[1,1,0,1,0,1,0,0] generating graphics... => 2
[1,1,0,1,1,0,0,0] generating graphics... => 1
[1,1,1,0,0,0,1,0] generating graphics... => 3
[1,1,1,0,0,1,0,0] generating graphics... => 2
[1,1,1,0,1,0,0,0] generating graphics... => 1
[1,1,1,1,0,0,0,0] generating graphics... => 0
[1,0,1,0,1,0,1,0,1,0] generating graphics... => 10
[1,0,1,0,1,0,1,1,0,0] generating graphics... => 6
[1,0,1,0,1,1,0,0,1,0] generating graphics... => 7
[1,0,1,0,1,1,0,1,0,0] generating graphics... => 6
[1,0,1,0,1,1,1,0,0,0] generating graphics... => 3
[1,0,1,1,0,0,1,0,1,0] generating graphics... => 8
[1,0,1,1,0,0,1,1,0,0] generating graphics... => 4
[1,0,1,1,0,1,0,0,1,0] generating graphics... => 7
[1,0,1,1,0,1,0,1,0,0] generating graphics... => 4
[1,0,1,1,0,1,1,0,0,0] generating graphics... => 3
[1,0,1,1,1,0,0,0,1,0] generating graphics... => 5
[1,0,1,1,1,0,0,1,0,0] generating graphics... => 4
[1,0,1,1,1,0,1,0,0,0] generating graphics... => 3
[1,0,1,1,1,1,0,0,0,0] generating graphics... => 1
[1,1,0,0,1,0,1,0,1,0] generating graphics... => 9
[1,1,0,0,1,0,1,1,0,0] generating graphics... => 5
[1,1,0,0,1,1,0,0,1,0] generating graphics... => 6
[1,1,0,0,1,1,0,1,0,0] generating graphics... => 5
[1,1,0,0,1,1,1,0,0,0] generating graphics... => 2
[1,1,0,1,0,0,1,0,1,0] generating graphics... => 8
[1,1,0,1,0,0,1,1,0,0] generating graphics... => 4
[1,1,0,1,0,1,0,0,1,0] generating graphics... => 6
[1,1,0,1,0,1,0,1,0,0] generating graphics... => 4
[1,1,0,1,0,1,1,0,0,0] generating graphics... => 2
[1,1,0,1,1,0,0,0,1,0] generating graphics... => 5
[1,1,0,1,1,0,0,1,0,0] generating graphics... => 4
[1,1,0,1,1,0,1,0,0,0] generating graphics... => 2
[1,1,0,1,1,1,0,0,0,0] generating graphics... => 1
[1,1,1,0,0,0,1,0,1,0] generating graphics... => 7
[1,1,1,0,0,0,1,1,0,0] generating graphics... => 3
[1,1,1,0,0,1,0,0,1,0] generating graphics... => 6
[1,1,1,0,0,1,0,1,0,0] generating graphics... => 3
[1,1,1,0,0,1,1,0,0,0] generating graphics... => 2
[1,1,1,0,1,0,0,0,1,0] generating graphics... => 5
[1,1,1,0,1,0,0,1,0,0] generating graphics... => 3
[1,1,1,0,1,0,1,0,0,0] generating graphics... => 2
[1,1,1,0,1,1,0,0,0,0] generating graphics... => 1
[1,1,1,1,0,0,0,0,1,0] generating graphics... => 4
[1,1,1,1,0,0,0,1,0,0] generating graphics... => 3
[1,1,1,1,0,0,1,0,0,0] generating graphics... => 2
[1,1,1,1,0,1,0,0,0,0] generating graphics... => 1
[1,1,1,1,1,0,0,0,0,0] generating graphics... => 0
[1,0,1,0,1,0,1,0,1,0,1,0] generating graphics... => 15
[1,0,1,0,1,0,1,0,1,1,0,0] generating graphics... => 10
[1,0,1,0,1,0,1,1,0,0,1,0] generating graphics... => 11
[1,0,1,0,1,0,1,1,0,1,0,0] generating graphics... => 10
[1,0,1,0,1,0,1,1,1,0,0,0] generating graphics... => 6
[1,0,1,0,1,1,0,0,1,0,1,0] generating graphics... => 12
[1,0,1,0,1,1,0,0,1,1,0,0] generating graphics... => 7
[1,0,1,0,1,1,0,1,0,0,1,0] generating graphics... => 11
[1,0,1,0,1,1,0,1,0,1,0,0] generating graphics... => 7
[1,0,1,0,1,1,0,1,1,0,0,0] generating graphics... => 6
[1,0,1,0,1,1,1,0,0,0,1,0] generating graphics... => 8
[1,0,1,0,1,1,1,0,0,1,0,0] generating graphics... => 7
[1,0,1,0,1,1,1,0,1,0,0,0] generating graphics... => 6
[1,0,1,0,1,1,1,1,0,0,0,0] generating graphics... => 3
[1,0,1,1,0,0,1,0,1,0,1,0] generating graphics... => 13
[1,0,1,1,0,0,1,0,1,1,0,0] generating graphics... => 8
[1,0,1,1,0,0,1,1,0,0,1,0] generating graphics... => 9
[1,0,1,1,0,0,1,1,0,1,0,0] generating graphics... => 8
[1,0,1,1,0,0,1,1,1,0,0,0] generating graphics... => 4
[1,0,1,1,0,1,0,0,1,0,1,0] generating graphics... => 12
[1,0,1,1,0,1,0,0,1,1,0,0] generating graphics... => 7
[1,0,1,1,0,1,0,1,0,0,1,0] generating graphics... => 9
[1,0,1,1,0,1,0,1,0,1,0,0] generating graphics... => 7
[1,0,1,1,0,1,0,1,1,0,0,0] generating graphics... => 4
[1,0,1,1,0,1,1,0,0,0,1,0] generating graphics... => 8
[1,0,1,1,0,1,1,0,0,1,0,0] generating graphics... => 7
[1,0,1,1,0,1,1,0,1,0,0,0] generating graphics... => 4
[1,0,1,1,0,1,1,1,0,0,0,0] generating graphics... => 3
[1,0,1,1,1,0,0,0,1,0,1,0] generating graphics... => 10
[1,0,1,1,1,0,0,0,1,1,0,0] generating graphics... => 5
[1,0,1,1,1,0,0,1,0,0,1,0] generating graphics... => 9
[1,0,1,1,1,0,0,1,0,1,0,0] generating graphics... => 5
[1,0,1,1,1,0,0,1,1,0,0,0] generating graphics... => 4
[1,0,1,1,1,0,1,0,0,0,1,0] generating graphics... => 8
[1,0,1,1,1,0,1,0,0,1,0,0] generating graphics... => 5
[1,0,1,1,1,0,1,0,1,0,0,0] generating graphics... => 4
[1,0,1,1,1,0,1,1,0,0,0,0] generating graphics... => 3
[1,0,1,1,1,1,0,0,0,0,1,0] generating graphics... => 6
[1,0,1,1,1,1,0,0,0,1,0,0] generating graphics... => 5
[1,0,1,1,1,1,0,0,1,0,0,0] generating graphics... => 4
[1,0,1,1,1,1,0,1,0,0,0,0] generating graphics... => 3
[1,0,1,1,1,1,1,0,0,0,0,0] generating graphics... => 1
[1,1,0,0,1,0,1,0,1,0,1,0] generating graphics... => 14
[1,1,0,0,1,0,1,0,1,1,0,0] generating graphics... => 9
[1,1,0,0,1,0,1,1,0,0,1,0] generating graphics... => 10
[1,1,0,0,1,0,1,1,0,1,0,0] generating graphics... => 9
[1,1,0,0,1,0,1,1,1,0,0,0] generating graphics... => 5
[1,1,0,0,1,1,0,0,1,0,1,0] generating graphics... => 11
[1,1,0,0,1,1,0,0,1,1,0,0] generating graphics... => 6
[1,1,0,0,1,1,0,1,0,0,1,0] generating graphics... => 10
[1,1,0,0,1,1,0,1,0,1,0,0] generating graphics... => 6
[1,1,0,0,1,1,0,1,1,0,0,0] generating graphics... => 5
[1,1,0,0,1,1,1,0,0,0,1,0] generating graphics... => 7
[1,1,0,0,1,1,1,0,0,1,0,0] generating graphics... => 6
[1,1,0,0,1,1,1,0,1,0,0,0] generating graphics... => 5
[1,1,0,0,1,1,1,1,0,0,0,0] generating graphics... => 2
[1,1,0,1,0,0,1,0,1,0,1,0] generating graphics... => 13
[1,1,0,1,0,0,1,0,1,1,0,0] generating graphics... => 8
[1,1,0,1,0,0,1,1,0,0,1,0] generating graphics... => 9
[1,1,0,1,0,0,1,1,0,1,0,0] generating graphics... => 8
[1,1,0,1,0,0,1,1,1,0,0,0] generating graphics... => 4
[1,1,0,1,0,1,0,0,1,0,1,0] generating graphics... => 11
[1,1,0,1,0,1,0,0,1,1,0,0] generating graphics... => 6
[1,1,0,1,0,1,0,1,0,0,1,0] generating graphics... => 9
[1,1,0,1,0,1,0,1,0,1,0,0] generating graphics... => 6
[1,1,0,1,0,1,0,1,1,0,0,0] generating graphics... => 4
[1,1,0,1,0,1,1,0,0,0,1,0] generating graphics... => 7
[1,1,0,1,0,1,1,0,0,1,0,0] generating graphics... => 6
[1,1,0,1,0,1,1,0,1,0,0,0] generating graphics... => 4
[1,1,0,1,0,1,1,1,0,0,0,0] generating graphics... => 2
[1,1,0,1,1,0,0,0,1,0,1,0] generating graphics... => 10
[1,1,0,1,1,0,0,0,1,1,0,0] generating graphics... => 5
[1,1,0,1,1,0,0,1,0,0,1,0] generating graphics... => 9
[1,1,0,1,1,0,0,1,0,1,0,0] generating graphics... => 5
[1,1,0,1,1,0,0,1,1,0,0,0] generating graphics... => 4
[1,1,0,1,1,0,1,0,0,0,1,0] generating graphics... => 7
[1,1,0,1,1,0,1,0,0,1,0,0] generating graphics... => 5
[1,1,0,1,1,0,1,0,1,0,0,0] generating graphics... => 4
[1,1,0,1,1,0,1,1,0,0,0,0] generating graphics... => 2
[1,1,0,1,1,1,0,0,0,0,1,0] generating graphics... => 6
[1,1,0,1,1,1,0,0,0,1,0,0] generating graphics... => 5
[1,1,0,1,1,1,0,0,1,0,0,0] generating graphics... => 4
[1,1,0,1,1,1,0,1,0,0,0,0] generating graphics... => 2
[1,1,0,1,1,1,1,0,0,0,0,0] generating graphics... => 1
[1,1,1,0,0,0,1,0,1,0,1,0] generating graphics... => 12
[1,1,1,0,0,0,1,0,1,1,0,0] generating graphics... => 7
[1,1,1,0,0,0,1,1,0,0,1,0] generating graphics... => 8
[1,1,1,0,0,0,1,1,0,1,0,0] generating graphics... => 7
[1,1,1,0,0,0,1,1,1,0,0,0] generating graphics... => 3
[1,1,1,0,0,1,0,0,1,0,1,0] generating graphics... => 11
[1,1,1,0,0,1,0,0,1,1,0,0] generating graphics... => 6
[1,1,1,0,0,1,0,1,0,0,1,0] generating graphics... => 8
[1,1,1,0,0,1,0,1,0,1,0,0] generating graphics... => 6
[1,1,1,0,0,1,0,1,1,0,0,0] generating graphics... => 3
[1,1,1,0,0,1,1,0,0,0,1,0] generating graphics... => 7
[1,1,1,0,0,1,1,0,0,1,0,0] generating graphics... => 6
[1,1,1,0,0,1,1,0,1,0,0,0] generating graphics... => 3
[1,1,1,0,0,1,1,1,0,0,0,0] generating graphics... => 2
[1,1,1,0,1,0,0,0,1,0,1,0] generating graphics... => 10
[1,1,1,0,1,0,0,0,1,1,0,0] generating graphics... => 5
[1,1,1,0,1,0,0,1,0,0,1,0] generating graphics... => 8
[1,1,1,0,1,0,0,1,0,1,0,0] generating graphics... => 5
[1,1,1,0,1,0,0,1,1,0,0,0] generating graphics... => 3
[1,1,1,0,1,0,1,0,0,0,1,0] generating graphics... => 7
[1,1,1,0,1,0,1,0,0,1,0,0] generating graphics... => 5
[1,1,1,0,1,0,1,0,1,0,0,0] generating graphics... => 3
[1,1,1,0,1,0,1,1,0,0,0,0] generating graphics... => 2
[1,1,1,0,1,1,0,0,0,0,1,0] generating graphics... => 6
[1,1,1,0,1,1,0,0,0,1,0,0] generating graphics... => 5
[1,1,1,0,1,1,0,0,1,0,0,0] generating graphics... => 3
[1,1,1,0,1,1,0,1,0,0,0,0] generating graphics... => 2
[1,1,1,0,1,1,1,0,0,0,0,0] generating graphics... => 1
[1,1,1,1,0,0,0,0,1,0,1,0] generating graphics... => 9
[1,1,1,1,0,0,0,0,1,1,0,0] generating graphics... => 4
[1,1,1,1,0,0,0,1,0,0,1,0] generating graphics... => 8
[1,1,1,1,0,0,0,1,0,1,0,0] generating graphics... => 4
[1,1,1,1,0,0,0,1,1,0,0,0] generating graphics... => 3
[1,1,1,1,0,0,1,0,0,0,1,0] generating graphics... => 7
[1,1,1,1,0,0,1,0,0,1,0,0] generating graphics... => 4
[1,1,1,1,0,0,1,0,1,0,0,0] generating graphics... => 3
[1,1,1,1,0,0,1,1,0,0,0,0] generating graphics... => 2
[1,1,1,1,0,1,0,0,0,0,1,0] generating graphics... => 6
[1,1,1,1,0,1,0,0,0,1,0,0] generating graphics... => 4
[1,1,1,1,0,1,0,0,1,0,0,0] generating graphics... => 3
[1,1,1,1,0,1,0,1,0,0,0,0] generating graphics... => 2
[1,1,1,1,0,1,1,0,0,0,0,0] generating graphics... => 1
[1,1,1,1,1,0,0,0,0,0,1,0] generating graphics... => 5
[1,1,1,1,1,0,0,0,0,1,0,0] generating graphics... => 4
[1,1,1,1,1,0,0,0,1,0,0,0] generating graphics... => 3
[1,1,1,1,1,0,0,1,0,0,0,0] generating graphics... => 2
[1,1,1,1,1,0,1,0,0,0,0,0] generating graphics... => 1
[1,1,1,1,1,1,0,0,0,0,0,0] generating graphics... => 0
click to show generating function       
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.
Code
def statistic(x):
    return x.bounce()
Created
Sep 15, 2011 at 15:56 by Chris Berg
Updated
Jun 17, 2019 at 17:01 by Christian Stump