A Reciprocity Theorem for Monomer-Dimer Coverings
| dc.creator | Anzalone, N. | |
| dc.creator | Baldwin, J. | |
| dc.creator | Bronshtein, I. | |
| dc.creator | Petersen, T. K. | |
| dc.date | 2003-04-23 | |
| dc.date | 2003-05-20 | |
| dc.date.accessioned | 2026-07-07T04:57:17Z | |
| dc.date.available | 2026-07-07T04:57:17Z | |
| dc.description | 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 $N(m,-2-n) = ε_{m,n}N(m,n)$ where $ε_{m,n} = 1$ unless $m\equiv$ 2(mod 4) and $n$ is odd, in which case $ε_{m,n} = -1$. 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. | |
| dc.description | 13 pages, 12 figures, to appear in the proceedings of the Discrete Models for Complex Systems (DMCS) 2003 conference. (v2 - some minor changes) | |
| dc.identifier | https://arxiv.org/abs/math/0304359 | |
| dc.identifier | http://arxiv.org/abs/math/0304359 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/67213 | |
| dc.subject | Combinatorics | |
| dc.subject | 05A15; 05C22; 05C30; 05C70 | |
| dc.title | A Reciprocity Theorem for Monomer-Dimer Coverings | |
| dc.type | text |