A Reciprocity Theorem for Monomer-Dimer Coverings

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 ((Kas61), (TF61)). In earlier work, Stanley (Sta85) 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 (Pro01), Stanley’s result concerns the unique way of extending N(m,n) to n < 0 so that the resulting biinfinite 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,nN(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 was applicable to higherdimensional 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.