Two combinatorial statistics on Dyck paths Original Research Article

Two combinatorial statistics, the pyramid weight and the number of exterior pairs, are investigated on the set of Dyck paths. Explicit formulae are given for the generating functions of Dyek paths of prescribed pyramid weight and prescribed number of exterior pairs. The proofs are combinatorial and rely on the method of q-grammars as well as on two new q-analogues of the Catalan numbers derived from statistics on non-crossing partitions. Connections with the combinatories of Motzkin paths are pointed out.