Counting cluster-tilted algebras of type $A_n$

TL;DR

This paper provides an explicit formula for counting non-isomorphic cluster-tilted algebras of type A_n by analyzing mutation classes of quivers, and characterizes when endomorphism algebras are isomorphic.

Contribution

It introduces a formula for enumerating cluster-tilted algebras of type A_n and characterizes isomorphisms of their endomorphism algebras in cluster categories.

Findings

Derived an explicit counting formula for cluster-tilted algebras of type A_n.

Established conditions for isomorphism of endomorphism algebras in cluster categories.

Connected algebraic structures with mutation class enumeration.

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 $T'$ 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 $T=\tau^i T'$.