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

TL;DR

This paper explores the combinatorial structures of snake graphs, establishing bijections with lattice paths and tilings, and deriving formulas for perfect matchings using determinants and Fibonacci-related sequences.

Contribution

It introduces triangular snake graphs and links their perfect matchings to lattice paths and tilings, providing new combinatorial formulas involving determinants and Fibonacci numbers.

Findings

Number of perfect matchings expressed via Hankel determinants with Catalan numbers.

Perfect matchings counted as sums of products of Fibonacci numbers.

Fibonacci and Pell sequences derived from determinants of Fibonacci matrices.

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.