Counting cluster-tilted algebras of type $A_n$

Abstract

The purpose of this paper is to give an explicit formula for the number of non-isomorphic cluster-tilted algebras of type $A_n$, by counting the mutation class of any quiver with underlying graph $A_n$. It will also follow that if $T$ and $T^{\prime }$ are cluster-tilting objects in a cluster category $\mathcal{C}$, then $\End_{\mathcal{C}}(T)$ is isomorphic to $\End_{\mathcal{C}}(T')$ if and only if $T={\tau }^i{T}^{\prime }$.

9 pages, 4 figures, minor changes, grammatical corrections and layout