General Results on the Enumeration of Strings in Dyck Paths

Let τ 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 τ” has been studied by many authors, for particular strings only. In this paper, arbitrary strings are considered. The associated generating function is evaluated when τ is a Dyck prefix (or a Dyck suffix). Furthermore, the case when τ is neither a Dyck prefix nor a Dyck suffix is considered, giving some partial results. Finally, the statistic “number of occurrences of τ at height at least j” is considered, evaluating the corresponding generating function when τ is either a Dyck prefix or a Dyck suffix.