Enumeration of Dyck Paths with Air Pockets
Jean-Luc Baril, Sergey Kirgizov, Rémi Maréchal, and
Université de Bourgogne
B.P. 47 870
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.
Full version: pdf,
(Concerned with sequences
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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