New Combinatorial Formulas for Cluster Monomials of Type A Quivers

Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type A cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial formula for the cluster monomials in terms of the so-called globally compatible collections. We give bijective proofs of these formulas by comparing with the well-known combinatorial models of the T -paths and of the perfect matchings in a snake diagram. For cluster variables of a type A cluster algebra, we give a bijection that relates our new formula with the theta functions constructed by Gross, Hacking, Keel and Kontsevich. ∗Supported by the Korea Institute for Advanced Study (KIAS), the AMS Centennial Fellowship, NSA grant H98230-14-1-0323, and the University of Nebraska–Lincoln. †Supported by the Oakland University URC Faculty Research Fellowship Award, and NSA grant H98230-16-1-0303. the electronic journal of combinatorics 24(2) (2017), #P2.42 1