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 $\sum_{d|n} \phi(n/d)\binom{2d}{d}/(2n)$ for $n \geq 5$. To obtain this formula, we give a correspondence between the quivers in the mutation class and certain rooted trees.