Combinatorial connections in snake graphs: Tilings, lattice paths, and perfect matchings

Abstract

Snake graphs and their perfect matchings play a key role in the description of cluster variables of cluster algebras associated to surfaces. In this paper, we introduce triangular snake graphs and establish a bijection between their routes (non-intersecting lattice paths), perfect matchings of their underlying snake graphs, and tilings. As an application, we show that the number of perfect matchings in straight snake graphs can be expressed in terms of determinants of Hankel matrices with Catalan number entries. Moreover, we prove that the number of perfect matchings in snake graphs can be expressed as a sum of products of Fibonacci numbers, and we show how Fibonacci and Pell sequences arise from determinants of matrices with Fibonacci entries.