Counting strings at height j in Dyck paths

Let τ be an arbitrary lattice path, called in this context string, consisting of two kinds of steps (rises and falls) and let j be a non-negative integer. s paper, the explicit formula for the generating function Fj associated with the Dyck path statistic “number of occurrences of τ at height j” is evaluated. e expression of Fj some basic characteristics of the string are used, namely its number of rises, height, depth and periodicity, as well as the generating function of the Catalan numbers.