Left to right maxima in Dyck prefixes

Abstract

In a Dyck path, a peak which is strictly (weakly) higher than all the preceding peaks is called a strict (weak) left-to-right maximum. By dropping the restrictions for the path to end on the $x$-axis, one obtains Dyck prefixes.
We obtain explicit generating functions for both weak and strict left-to-right maxima in Dyck prefixes.
The proofs of the associated asymptotics make use of analytic techniques such as Mellin transforms, singularity analysis and formal residue calculus.