A Reciprocity Theorem for Monomer-Dimer Coverings

Abstract

The problem of counting monomer-dimer coverings of a lattice is a longstanding problem in statistical mechanics.It has only been exactly solved for the special case of dimer coverings in two dimensions ([Ka61], [TF61]). In earlier work, Stanley [St85] proved a reciprocity principle governing the number $N(m,n)$ of dimer coverings of an $m$ by $n$ rectangular grid (also known as perfect matchings), where $m$ is fixed and $n$ is allowed to vary. As reinterpreted by Propp [P01], Stanley's result concerns the unique way of extending $N(m,n)$ to $n<0$ so that the resulting bi-infinite sequence, $N(m,n)$ for $n \in \mathbb{Z}$, satisfies a linear recurrence relation with constant coefficients. In particular, Stanley shows that $N(m,n)$ is always an integer satisfying the relation $N(m,-2-n) = \varepsilon_{m,n} N(m,n)$ where $\varepsilon_{m,n}=1$ unless $m \equiv 2(\mod 4)$ and $n$ is odd, in which case $\varepsilon_{m,n}=-1$. Furthermore, Propp's method was applicable to higher-dimensional cases.This paper discusses similar investigations of the numbers $M(m,n)$, of monomer-dimer coverings, or equivalently (not necessarily perfect) matchings of an $m$ by $n$ rectangular grid. We show that for each fixed $m$ there is a unique way of extending $M(m,n)$ to $n<0$ so that the resulting bi-infinite sequence, $M(m,n)$ for $n \in \mathbb{Z}$, satisfies a linear recurrence relation with constant coefficients.We show that $M(m,n)$, a priori a rational number, is always an integer, using a generalization of the combinatorial model offered by Propp. Lastly, we give a new statement of reciprocity in terms of multivariate generating functions from which Stanley's result follows.