Equivalence classes of Dyck paths modulo some statistics

We investigate new equivalence relations on the set DD of Dyck paths relatively to the three statistics of double rises, peaks and valleys. Two Dyck paths are rr-equivalent (respectively pp-equivalent and vv-equivalent) whenever the positions of their double rises (respectively peaks and valleys) are the same. Then, we provide generating functions for the numbers of rr-, pp- and vv-equivalence classes of DD.