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

TL;DR

The paper develops a combinatorial bridge between snake graphs, tilings, and non-intersecting lattice paths by introducing triangular snake graphs and a bijection between perfect matchings, $k$-routes, and tilings. Using edge contractions and the Lindström–Gessel–Viennot framework, it connects these objects to Hankel determinants built from Catalan, Fibonacci, and Pell numbers, yielding exact determinant identities such as $\det M_{st}=F_{2k+1}$ for ladder-type graphs and $\det(H_k(C))+\det(H'_k(C))=F_{2k+1}$. A key outcome is that the number of perfect matchings in snake graphs can be expressed as sums of products of Fibonacci numbers, and certain path-matrix determinants reproduce Pell numbers, thereby unifying tilings, paths, and classical integer sequences within a cluster-algebra context. These results offer new combinatorial interpretations of continued fractions associated with snake graphs and suggest broader implications for the algebraic structure of cluster variables on surfaces.

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.

Figures (9)

• Figure 1: From left to right, tile $G_i$ of a snake graph $\mathcal{G}$ and its cover $T(G_i)$.
• Figure 2: Decorated dominoes.
• Figure 3: Triangular tiles associated to the tiles $G_d$ and $G_{d+1}$ that share the north and south edge.
• Figure 4: Triangular tiles associated to the tiles $G_d$ and $G_{d+1}$ that share the east and west edge.
• Figure 5: The 3 cases in which $p$ is a path starting at $s_i$ and ending at $t_j$, $i\neq j$.

Theorems & Definitions (59)

• Theorem 1
• Theorem 2
• Lemma 3: Lindström Lin73, Gessel-Viennot GV89
• Theorem 4: CS18
• Theorem 5: CS18
• Definition 6
• Definition 7
• Example 8
• Proposition 9
• proof