Abstract
In the present paper, we consider the statistic “number of udu's” 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 udu's” has been considered only recently. Let D n denote the set of Dyck paths of semilength n and let T n , k , L n , k , H n , k and W n , k ( r ) denote the number of Dyck paths in D n with k udu's, with k udu's at low level, at high level, and at level r ⩾ 2 , 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.
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