"Cyclic sieving", introduced by Reiner, Stanton and White in 2004, is an enumerative property of a set of combinatorial objects on which a cyclic group acts. We will discuss recent results on the cyclic sieving phenomenon for embedded graphs.
Our results include first families of trees, in a paper in collaboration with Christian Krattenthaler (Vienna), and then families of planar maps, in collaboration with Marie Albenque (Paris).