  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.
