Identifier
Values
[1] => 10 => [1,2] => [1,0,1,1,0,0] => 2
[2] => 100 => [1,3] => [1,0,1,1,1,0,0,0] => 3
[1,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 2
[3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 4
[2,1] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 4
[1,1,1] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 2
[4] => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[3,1] => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 6
[2,2] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
[2,1,1] => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 4
[1,1,1,1] => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
[3,2] => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 6
[2,2,1] => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 4
[3,3] => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 4
[2,2,2] => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
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Description
The product of the heights of the peaks of a Dyck path.
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 partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.
Map
bounce path
Description
The bounce path determined by an integer composition.