Counting the Number of Elements in the Mutation Classes of $\tilde A_n-$Quivers

Abstract

In this article we prove explicit formulae for the number of non-isomorphic cluster-tilted algebras of type $\tilde A_n$ in the derived equivalence classes. In particular, we obtain the number of elements in the mutation classes of quivers of type $\tilde A_n$. As a by-product, this provides an alternative proof for the number of quivers mutation equivalent to a quiver of Dynkin type $D_n$ which was first determined by Buan and Torkildsen.