The Number of Elements in the Mutation Class of a Quiver of Type $D_n$

Abstract

We show that the number of quivers in the mutation class of a quiver of Dynkin type $D_n$ is given by $\sum_{d|n}\phi(n/d){2d\choose d}/(2n)$ for $n \geq 5$. To obtain this formula, we give a correspondence between the quivers in the mutation class and certain rooted trees.