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∈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)=ϵm,n​N(m,n) where ϵm,n​=1 unless m≡ 2(mod 4) and
n is odd, in which case ϵm,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∈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.Comment: 13 pages, 12 figures, to appear in the proceedings of the Discrete
Models for Complex Systems (DMCS) 2003 conference. (v2 - some minor changes