Counting strings at height j in Dyck paths

Let ττ 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.In this paper, the explicit formula for the generating function Fj associated with the Dyck path statistic “number of occurrences of ττ at height j” is evaluated.For the 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.