Higher Dimer Covers on Snake Graphs

TL;DR

This paper extends the classical bridge between snake graphs and continued fractions by counting $m$-dimer covers, introducing $(m+1)\times(m+1)$-dimensional matrices $\Lambda^{(m)}(a)$ and showing that the number of $m$-dimer covers of a snake graph $\mathscr{G}[a_1,\dots,a_n]$ equals the top-left entry of $\Lambda^{(m)}(a_1)\cdots\Lambda^{(m)}(a_n)$. It develops a rich algebraic framework linking these counts to generalized continued fractions $\mathrm{CF}_m$, dual snake graphs, and generating functions via poset methods and quasi-symmetric functions, with special attention to $m=1$ recovering classical results and $m=2$ connecting to ternary continued fractions and cubic irrationals. The paper further explores the SL$_{m+1}(\mathbb{Z})$-structure behind the matrix products, asymptotic behavior of the generalized fractions, and potential connections to Hermite’s problem, while outlining numerous open questions and directions for future research. Overall, the work provides a unified combinatorial and algebraic approach to higher dimer enumerations, generalized continued fractions, and their number-theoretic implications. The results open avenues for q-analogues, unimodality questions, and inverse problems that may yield new insights into irrationality and 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.

Figures (7)

• Figure 1: The left hand picture shows the snake graph of the continued fraction $[2,4,1,2]$. The sign sequence is given in red. The sign changes occur in the tiles $G_{\ell_i}$, with $\ell_i=2,6,7$. The picture on the right shows the dual snake graph $(\mathcal{G}[2,4,1,2])^*=\mathcal{G}[1,2,1,1,3,1]$.
• Figure 2: The pictures in the top row show the five 1-dimer covers $P_1,\ldots,P_5$ of the snake graph $\mathcal{G}[1,1,1,1]$. The pictures in the bottom row show the five lattice paths $w_1,\ldots,w_5$ of $\mathcal{G}[4]$, which is the dual of $\mathcal{G}[1,1,1,1]$. The top picture on the right hand side shows that the multisets $\{P_2,P_5\}$ and $\{P_3,P_4\}$ produce the same 2-dimer cover, and the picture below shows that the multipaths $\{w_2,w_5\}$ and $\{w_3,w_4\}$ produce the same 2-lattice path.
• Figure 3: Bijection between dimer covers of $\mathcal{G}$ and lattice paths of $\mathcal{G}^*$.
• Figure 4: Poset structure of $\Omega_2[4]$
• Figure 5: The linear extensions of a fence poset. The labels next to the $b_i$'s are the values of the distinguished linear extension $\pi$.

Theorems & Definitions (80)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Theorem 2.2
• Example 2.3
• Remark 2.4
• Remark 2.5
• Theorem 2.6
• Proposition 3.1