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.
[1]
R. Stanley.
What Is Enumerative Combinatorics
,
1986
.
[2]
Richard P. Stanley,et al.
On dimer coverings of rectangles of fixed width
,
1985,
Discret. Appl. Math..
[3]
P. W. Kasteleyn.
The Statistics of Dimers on a Lattice
,
1961
.
[4]
M. Fisher,et al.
Dimer problem in statistical mechanics-an exact result
,
1961
.
[5]
Christian Krattenthaler,et al.
Combinatorial Proof of the Log-Concavity of the Sequence of Matching Numbers
,
1996,
J. Comb. Theory, Ser. A.
[6]
O. J. Heilmann,et al.
Theory of monomer-dimer systems
,
1972
.
[7]
James Gary Propp.
A Reciprocity Theorem for Domino Tilings
,
2001,
Electron. J. Comb..
[8]
A. Sinclair,et al.
Approximating the number of monomer-dimer coverings of a lattice
,
1996
.
[9]
David G. Wagner,et al.
Homogeneous multivariate polynomials with the half-plane property
,
2004,
Adv. Appl. Math..