Dyck paths with first sojourn highest
DOI:
https://doi.org/10.55016/ojs/cdm.v20i2.74008Abstract
A Dyck path is a lattice path in the first quadrant using steps $u=(1,1)$ and $d=(1,-1)$, starting at the origin and ending on the $x$-axis. A Dyck sojourn is a Dyck path with only one return to the $x$-axis. Various subclasses of Dyck paths are shown to be counted by Catalan numbers: Dyck paths with weakly highest first sojourn and semilength $n+1$ are counted by twice the $n$th Catalan number and Dyck paths with semilength $n+1$ where the second sojourn is weakly highest are counted by the $n$th Catalan number. Also Dyck paths in which the first sojourn is strictly highest are equinumerous to Dyck paths with exactly one peak of maximum height. Some of these equivalences are proved using bijections. Generating functions for all these subclasses are also developed in the paper. The asymptotics as $n\to\infty$ for the case where the first sojourn is strictly highest are explored.
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