Identifier
Values
[1] => [1,0,1,0] => [1,1,0,1,0,0] => 110100 => 2
[2] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 11100100 => 3
[1,1] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => 11011000 => 3
[2,1] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => 11010100 => 5
[] => [] => [1,0] => 10 => 1
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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
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.