A Cluster Expansion Formula ($A_n$ case) - Schiffler, Ralf (The Electronic Journal of Combinatorics, 2008)
We consider the Ptolemy cluster algebras, which are cluster algebras of finite type $A$ (with non-trivial coefficients) that have been described by Fomin and Zelevinsky using triangulations of a regular polygon. Given any seed $Σ$ in a Ptolemy cluster algebra, we present a formula for the expansion of an arbitrary cluster variable in terms of the cluster variables of the seed $Σ$. Our formula is given in a combinatorial way, using paths on a triangulation of the polygon that corresponds to the seed $Σ$.