Higher dimer covers on snake graphs

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝒢</mml:mi> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>...</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> is the numerator of the continued fraction <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>...</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> . This number is equal to the top left entry of the matrix product <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced close=")" open="("> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtable> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mtd> <mml:mtd> <mml:mn>1</mml:mn> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mn>1</mml:mn> </mml:mtd> <mml:mtd> <mml:mn>0</mml:mn> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mstyle> </mml:mfenced> <mml:mo>⋯</mml:mo> <mml:mfenced close=")" open="("> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mtable> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mtd> <mml:mtd> <mml:mn>1</mml:mn> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mn>1</mml:mn> </mml:mtd> <mml:mtd> <mml:mn>0</mml:mn> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mstyle> </mml:mfenced> </mml:mrow> </mml:math> . In this paper, we give enumerative results on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> </mml:math> -dimer covers of snake graphs. We show that the number of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> </mml:math> -dimer covers of the snake graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝒢</mml:mi> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>...</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> is the top left entry of a product of analogous <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -by- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> 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.