Enumeration of Dyck Paths with Air Pockets

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. Finally, we conduct a similar study for non-decreasing Dyck paths with air pockets.