## DMTCS Proceedings, Discrete Models for Complex Systems, DMCS'03

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DMTCS Conference vol AB (2003), pp. 179-194

## Discrete Models for Complex Systems, DMCS'03

### DMTCS Conference Volume AB (2003), pp. 179-194

author: Nick Anzalone, John Baldwin, Ilya Bronshtein and T. Kyle Petersen A Reciprocity Theorem for Monomer-Dimer Coverings reciprocity, monomer-dimer coverings, linear recurrences 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 ZZ, 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 N(m,n) where epsilon = 1 unless m = 2(mod 4) and n is odd, in which case epsilon = -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 ZZ, 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. If your browser does not display the abstract correctly (because of the different mathematical symbols) you may look it up in the PostScript or PDF files. Nick Anzalone and John Baldwin and Ilya Bronshtein and T. Kyle Petersen (2003), A Reciprocity Theorem for Monomer-Dimer Coverings, in Discrete Models for Complex Systems, DMCS'03, Michel Morvan and Éric Rémila (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AB, pp. 179-194 For a corresponding BibTeX entry, please consider our BibTeX-file. dmAB0115.ps.gz (252 K) dmAB0115.ps (836 K) dmAB0115.pdf (196 K)