Identifier
Values
[1,0] => 10 => [1,2] => [1,0,1,1,0,0] => 1
[1,0,1,0] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,1,0,0] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
[1,0,1,0,1,0] => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 3
[1,0,1,1,0,0] => 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => 4
[1,1,0,0,1,0] => 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 4
[1,1,0,1,0,0] => 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => 4
[1,1,1,0,0,0] => 111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 6
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Description
The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps.
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.
Map
to binary word
Description
Return the Dyck word as binary word.