Enumeration of Dyck paths with air pockets

TL;DR

This paper introduces Dyck paths with air pockets, establishes a bijection with peakless Motzkin paths, and analyzes pattern distributions, asymptotics, and connections to Fibonacci meanders.

Contribution

It presents a new combinatorial class of Dyck paths with air pockets, along with bijections, generating functions, and asymptotic analysis, linking to Fibonacci meanders.

Findings

Bijection with peakless Motzkin paths established

Explicit bivariate generating functions derived

Asymptotic expectations of patterns computed

Abstract

We introduce and study the new combinatorial class of Dyck paths with air pockets. We exhibit a bijection with the peakless Motzkin paths which transports several pattern statistics and give bivariate generating functions for the distribution of patterns as peaks, returns and pyramids. Then, we deduce the popularities and asymptotic expectations of these patterns and point out a link between the popularity of pyramids and a special kind of closed smooth self-overlapping curves, a subset of Fibonacci meanders. A similar study is conducted for non-decreasing Dyck paths with air pockets.