A cluster expansion formula (An case)

Abstract

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.