Riordan paths and derangements

Riordan paths are Motzkin paths without horizontal steps on the x  -axis. We establish a correspondence between Riordan paths and (321,31¯42)-avoiding derangements. We also present a combinatorial proof of a recurrence relation for the Riordan numbers in the spirit of the Foata–Zeilberger proof of a recurrence relation on the Schröder numbers.