Identifier
Values
[1,0] => [1,0] => 10 => 1
[1,0,1,0] => [1,1,0,0] => 1100 => 1
[1,1,0,0] => [1,0,1,0] => 1010 => 2
[1,0,1,0,1,0] => [1,1,0,0,1,0] => 110010 => 3
[1,0,1,1,0,0] => [1,1,0,1,0,0] => 110100 => 2
[1,1,0,0,1,0] => [1,1,1,0,0,0] => 111000 => 1
[1,1,0,1,0,0] => [1,0,1,1,0,0] => 101100 => 3
[1,1,1,0,0,0] => [1,0,1,0,1,0] => 101010 => 5
[1,0,1,0,1,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 6
[1,0,1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 3
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,0,1,0] => 11001010 => 9
[1,0,1,1,0,1,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 7
[1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => 5
[1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,0] => 11100100 => 3
[1,1,0,0,1,1,0,0] => [1,1,1,1,0,0,0,0] => 11110000 => 1
[1,1,0,1,0,0,1,0] => [1,1,1,0,0,0,1,0] => 11100010 => 4
[1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => 10
[1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 7
[1,1,1,0,0,0,1,0] => [1,1,1,0,1,0,0,0] => 11101000 => 2
[1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => 10111000 => 4
[1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 10101100 => 9
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => 14
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Description
The number of Dyck paths above the lattice path given by a binary word.
One may treat a binary word as a lattice path starting at the origin and treating $1$'s as steps $(1,0)$ and $0$'s as steps $(0,1)$. Given a binary word $w$, this statistic counts the number of lattice paths from the origin to the same endpoint as $w$ that stay weakly above $w$.
See St001312Number of parabolic noncrossing partitions indexed by the composition. for this statistic on compositions treated as bounce paths.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
Elizalde-Deutsch bijection
Description
The Elizalde-Deutsch bijection on Dyck paths.
.Let $n$ be the length of the Dyck path. Consider the steps $1,n,2,n-1,\dots$ of $D$. When considering the $i$-th step its corresponding matching step has not yet been read, let the $i$-th step of the image of $D$ be an up step, otherwise let it be a down step.