Journal of Integer Sequences, Vol. 26 (2023), Article 23.3.2

Enumeration of Dyck Paths with Air Pockets

Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and Vincent Vajnovszki
Université de Bourgogne
B.P. 47 870
21078 Dijon-Cedex


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.

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(Concerned with sequences A000108 A001006 A001519 A004148 A005043 A034839 A045891 A051291 A089732 A093128 A098086 A098156 A099036 A110320 A175657 A201631 A203611.)

Received September 6 2022; revised versions received March 3 2023, March 6 2023, March 8 2023, March 9 2023. Published in Journal of Integer Sequences, March 9 2023.

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