Higher dimer covers on snake graphs

Musiker, Gregg; Ovenhouse, Nicholas; Schiffler, Ralf; Zhang, Sylvester W

doi:10.5802/alco.464

Musiker, Gregg ¹; Ovenhouse, Nicholas ²; Schiffler, Ralf ³; Zhang, Sylvester W ⁴

¹University of Minnesota, Dept. of mathematics, 206 Church St. SE, Minneapolis, MN 55455 (USA)
²Michigan State University, Dept. of mathematics, 619 Red Cedar Rd, East Lansing, MI 48824 (USA)
³University of Connecticut, Dept. of mathematics, 341 Mansfield Rd U1009, Storrs, CT 06269-1009 (USA)
⁴UCLA, Dept. of mathematics, 520 Portola Plaza MS 6363, Los Angeles, CA 90095-1555 (USA)

Algebraic Combinatorics, Volume 9 (2026) no. 1, pp. 131-159

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 $\mathcal{G}[a_1,\ldots ,a_n]$ is the numerator of the continued fraction $[a_1,\ldots ,a_n]$. This number is equal to the top left entry of the matrix product $\left({{\textstyle \begin{matrix} a_1&1\\1&0 \end{matrix}}}\right) \cdots \left({{\textstyle \begin{matrix} a_n&1\\1&0 \end{matrix}}}\right)$.

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 $\mathcal{G}[a_1,\ldots ,a_n]$ 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.

Received: 2025-03-03
Revised: 2025-10-14
Accepted: 2025-10-18
Published online: 2026-03-03

DOI: 10.5802/alco.464

Classification: 05A15, 05C70, 11A55
Keywords: continued fractions, dimer covers, snake graphs

Author's affiliations:

Musiker, Gregg ¹ ; Ovenhouse, Nicholas ² ; Schiffler, Ralf ³ ; Zhang, Sylvester W ⁴

¹ University of Minnesota, Dept. of mathematics, 206 Church St. SE, Minneapolis, MN 55455 (USA)
² Michigan State University, Dept. of mathematics, 619 Red Cedar Rd, East Lansing, MI 48824 (USA)
³ University of Connecticut, Dept. of mathematics, 341 Mansfield Rd U1009, Storrs, CT 06269-1009 (USA)
⁴ UCLA, Dept. of mathematics, 520 Portola Plaza MS 6363, Los Angeles, CA 90095-1555 (USA)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Higher dimer covers on snake graphs},
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Musiker, Gregg; Ovenhouse, Nicholas; Schiffler, Ralf; Zhang, Sylvester W. Higher dimer covers on snake graphs. Algebraic Combinatorics, Volume 9 (2026) no. 1, pp. 131-159. doi: 10.5802/alco.464

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