Summary: In this paper we consider several statistics on the set of Dyck paths. Enumeration of Dyck paths according to length and various other parameters has been studied in several papers. However, the statistic "number of $udu$'s" has been considered only recently. We generalize this statistic and derive an explicit formula for the number of Dyck paths of length $2n$ according to the statistic "number of uu $\dots $udu's" ("number of udud$\dots $udu's"). As a consequence, we derive several known results, as well as many new results.

05A05, 05A15, 42C05

Chebyshev polynomials, Dyck paths, generating functions