The Cyclic Sieving Phenomenon and frieze patterns

We exhibit two instances of the cyclic sieving phenomenon - one on dissections of a polygon of a fixed type and one on triangulations of a once-punctured polygon. We use these results to give refined enumerations of certain families of frieze patterns. We also give an interpretation of finite, positive integral frieze patterns fixed under nontrivial rotations as frieze patterns from a family of orbifolds and show that these are always unitary. Finally, we give a bijection between Holm-Jorgensen frieze patterns and p-Dyck paths, extending a recent construction of Canadas, Espinosa, Gaviria, and Rios, and discuss an induced rotation map on Dyck paths. Several conjectures and questions for future study are highlighted throughout the article.