Symmetries in Dyck paths with air pockets

The main objective of this paper is to analyze symmetric and asymmetric peaks in Dyck paths with air pockets (DAPs). These paths are formed by combining each maximal run of down-steps in ordinary Dyck paths into a larger, single down-step. To achieve this, we present a trivariate generating function that counts the number of DAPs based on their length and the number of symmetric and asymmetric peaks they contain. We determine the total numbers of symmetric and asymmetric peaks across all DAPs, providing an asymptotic for the ratio of these two quantities. Recursive relations and closed formulas are provided for the number of DAPs of length n, as well as for the total number of symmetric peaks, weight of symmetric peaks, and height of symmetric peaks. Furthermore, a recursive relation is established for the overall number of DAPs, similar to that for classic Dyck paths. A DAP is said to be non-decreasing if the sequence of ordinates of all local minima forms a non-decreasing sequence. In the last section, we focus on the sets of non-decreasing DAPs and examine their symmetric and asymmetric peaks.