A Simple and Unusual Bijection for Dyck Paths and its Consequences

In this paper we introduce a new bijection from the set of Dyck paths to itself. This bijection has the property that it maps statistics that appeared recently in the study of patternavoiding permutations into classical statistics on Dyck paths, whose distribution is easy to obtain. We also present a generalization of the bijection, as well as several applications of it to enumeration problems of statistics in restricted permutations.AMS Subject Classification: 05A15, 05A05.