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.

翻译：我们引入并研究了具有气泡的Dyck路径这一新的组合结构类。我们展示了该类与无峰Motzkin路径之间的一个双射，该双射传递了若干模式统计量，并给出了峰、返回和金字塔等模式分布的二元生成函数。随后，我们推导了这些模式的流行度与渐近期望，并指出金字塔的流行度与一类特殊的封闭光滑自交曲线（斐波那契曲流的一个子集）之间的关联。我们进一步对具有气泡的非降Dyck路径进行了类似的研究。