We prove several new instances of the cyclic sieving phenomenon (CSP) on Catalan objects of type \(A\) and type \(B\). Moreover, we refine many of the known instances of the CSP on Catalan objects. For example, we consider triangulations refined by the number of "ears", non-crossing matchings with a fixed number of short edges, and non-crossing configurations with a fixed number of loops and edges.
Keywords: Dyck paths, cyclic sieving, Narayana numbers, major index, q-analog.
Mathematics Subject Classifications: 05E18, 05A19, 05A30