Higher Dimer Covers on Snake Graphs

TL;DR

This paper extends the enumeration of dimer covers on snake graphs to higher dimensions, linking these counts to matrix products and proposing a generalized continued fraction framework with implications for algebraic number theory.

Contribution

It introduces formulas for counting m-dimer covers of snake graphs using higher-dimensional matrix products and explores their connection to generalized continued fractions.

Findings

Number of m-dimer covers equals top-left entry of a product of (m+1)-by-(m+1) matrices.

Established relations between dimer cover enumeration and matrix algebra.

Proposed a new class of generalized continued fractions with potential applications in algebraic number theory.

Abstract

Snake graphs are a class of planar graphs that are important in the theory of cluster algebras. Indeed, the Laurent expansions of the cluster variables in cluster algebras from surfaces are given as weight generating functions for 1-dimer covers (or perfect matchings) of snake graphs. Moreover, the enumeration of 1-dimer covers of snake graphs provides a combinatorial interpretation of continued fractions. In particular, the number of 1-dimer covers of the snake graph $\mathscr{G}[a_1,\dots,a_n]$ is the numerator of the continued fraction $[a_1,\dots,a_n]$. This number is equal to the top left entry of the matrix product $\left(\begin{smallmatrix} a_1&1\\1&0 \end{smallmatrix}\right) \cdots \left(\begin{smallmatrix} a_n&1\\1&0 \end{smallmatrix}\right)$. In this paper, we give enumerative results on $m$-dimer covers of snake graphs. We show that the number of $m$-dimer covers of the snake graph $\mathscr{G}[a_1,\ldots,a_n]$ is the top left entry of a product of analogous $(m+1)$-by-$(m+1)$ matrices. We discuss how our enumerative results are related to other known combinatorial formulas, and we suggest a generalization of continued fractions based on our methods. These generalized continued fractions provide some interesting open questions and a possibly novel approach towards Hermite's problem for cubic irrationals.