Counting Cluster - Tilted

The purpose of this paper is to give an explicit formula for the number of non-isomorphic cluster-tilted algebras of type An, by counting the mutation class of any quiver with underlying graph An. It will also follow that if T and T ′ are cluster-tilting objects in a cluster category C, then EndC(T ) is isomorphic to EndC(T ) if and only if T = τ T . 1. Cluster-tilted algebras The cluster category was introduced independently in [7] for type An and in [2] for the general case. Let D(modH) be the bounded derived category of the finitely generated modules over a finite dimensional hereditary algebraH over a field K. In [2] the cluster category was defined as the orbit category C = D(modH)/τ[1], where τ is the Auslander-Reiten translation and [1] the suspension functor. The cluster-tilted algebras are the algebras of the form Γ = EndC(T ) , where T is a cluster-tilting object in C. See [3]. Let Q be a quiver with no multiple arrows, no loops and no oriented cycles of length two. Mutation of Q at vertex k is a quiver Q obtained from Q in the following way. (1) Add a vertex k. (2) If there is a path i → k → j, then if there is an arrow from j to i, remove this arrow. If there is no arrow from j to i, add an arrow from i to j. (3) For any vertex i replace all arrows from i to k with arrows from k to i, and replace all arrows from k to i with arrows from i to k. (4) Remove the vertex k. We say that a quiver Q is mutation equivalent to Q, if Q can be obtained from Q by a finite number of mutations. The mutation class of Q is all quivers mutation equivalent to Q. It is known from [11] that the mutation class of a Dynkin quiver Q is finite. If Γ is a cluster-tilted algebra, then we say that Γ is of type An if it arises from the cluster category of a path algebra of Dynkin type An. Let Q be a quiver of a cluster-tilted algebra Γ. From [4], it is known that if Q is obtained from Q by a finite number of mutations, then there is a cluster-tilted