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We give a $q$-enumeration of circular Dyck paths, which is a superset of theclassical Dyck paths enumerated by the Catalan numbers. These objects haverecently been studied by Alexandersson and Panova. Furthermore, we show thatthis $q$-analogue exhibits the cyclic sieving phenomenon under a natural actionof the cyclic group. The enumeration and cyclic sieving is generalized toM\"obius paths. We also discuss properties of a generalization of cyclicsieving, which we call subset cyclic sieving. Finally, we also introduce thenotion of Lyndon-like cyclic sieving that concerns special recursive propertiesof combinatorial objects exhibiting the cyclic sieving phenomenon.

The cyclic sieving phenomenon on circular Dyck paths