Identifier
Values
0 => [2] => [1,1,0,0] => 0
1 => [1,1] => [1,0,1,0] => 1
00 => [3] => [1,1,1,0,0,0] => 0
01 => [2,1] => [1,1,0,0,1,0] => 2
10 => [1,2] => [1,0,1,1,0,0] => 1
11 => [1,1,1] => [1,0,1,0,1,0] => 3
000 => [4] => [1,1,1,1,0,0,0,0] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0] => 3
010 => [2,2] => [1,1,0,0,1,1,0,0] => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0] => 5
100 => [1,3] => [1,0,1,1,1,0,0,0] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0] => 4
110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 3
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0] => 6
0000 => [5] => [1,1,1,1,1,0,0,0,0,0] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0] => 4
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0] => 3
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 7
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0] => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 6
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 5
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 9
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 5
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 8
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 7
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 6
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 10
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 5
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 4
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 9
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 3
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 8
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 7
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 12
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 7
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 6
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 11
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 5
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 10
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 9
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 14
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 6
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 5
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 10
10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 4
10101 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 9
10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 8
10111 => [1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 13
11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 3
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 8
11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 7
11011 => [1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 12
11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 6
11101 => [1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 11
11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 10
11111 => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 15
=> [1] => [1,0] => 0
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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.
Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.