Counting strings in Dyck paths

This paper deals with the enumeration of Dyck paths according to the statistic “number of occurrences of ττ”, for an arbitrary string ττ. In this direction, the statistic “number of occurrences of ττ at height jj” is considered. It is shown that the corresponding generating function can be evaluated with the aid of Chebyshev polynomials of the second kind. This is applied to every string of length 4. Further results are obtained for the statistic “number of occurrences of ττ at even (or odd) height”.