Left to right maxima in Dyck Paths

TL;DR

This paper derives explicit generating functions for weak and strict left-to-right maxima in Dyck paths, using advanced analytic techniques to analyze their asymptotic behavior.

Contribution

It provides new explicit generating functions for these maxima in Dyck paths and applies sophisticated analytic methods for asymptotic analysis.

Findings

Explicit generating functions for weak and strict maxima in Dyck paths

Asymptotic formulas derived using Mellin transforms and singularity analysis

Analytic techniques applied to combinatorial structures

Abstract

In a Dyck path a peak which is (weakly) higher than all the preceding peaks is called a strict (weak) left to right maximum. We obtain explicit generating functions for both weak and strict left to right maxima in Dyck paths. The proofs of the associated asymptotics make use of analytic techniques such as Mellin transforms, singularity analysis and formal residue calculus.