  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. In earlier work,
Stanley 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,
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 $n \in {Z}$, satisfies a
linear recurrence relation with constant coefficients. In particular, Stanley
shows that $N(m,n)$ is always an integer satisfying the relation $N(m,-2-n) =
\epsilon_{m,n}N(m,n)$ where $\epsilon_{m,n} = 1$ unless $m\equiv$ 2(mod 4) and
$n$ is odd, in which case $\epsilon_{m,n} = -1$. Furthermore, Propp's method is
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 $n
\in {Z}$, 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.
