General Results on the Enumeration of Strings in Dyck Paths

Let $\tau$ be a fixed lattice path (called in this context string) on the integer plane, consisting of two kinds of steps. The Dyck path statistic "number of occurrences of $\tau$" has been studied by many authors, for particular strings only. In this paper, arbitrary strings are considered. The associated generating function is evaluated when $\tau$ is a Dyck prefix (or a Dyck suffix). Furthermore, the case when $\tau$ is neither a Dyck prefix nor a Dyck suffix is considered, giving some partial results. Finally, the statistic "number of occurrences of $\tau$ at height at least $j$" is considered, evaluating the corresponding generating function when $\tau$ is either a Dyck prefix or a Dyck suffix.