The statistic “number of uduʹs” in Dyck paths

In the present paper, we consider the statistic “number of uduʹs” in Dyck paths. The enumeration of Dyck paths according to semilength and various other parameters has been studied in several papers. However, the statistic “number of uduʹs” has been considered only recently. Let image denote the set of Dyck paths of semilength n and let image image, image and image denote the number of Dyck paths in image with k uduʹs, with k uduʹs at low level, at high level, and at level image, respectively. We derive their generating functions, their recurrence relations and their explicit formulas. A new setting counted by Motzkin numbers is also obtained. Several combinatorial identities are given and other identities are conjectured.