# A Reciprocity Theorem for Monomer-Dimer Coverings

Abstract : 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.
Keywords :
Type de document :
Communication dans un congrès
Michel Morvan and Éric Rémila. Discrete Models for Complex Systems, DMCS'03, 2003, Lyon, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AB, Discrete Models for Complex Systems (DMCS'03), pp.179-194, 2003, DMTCS Proceedings
Domaine :

Littérature citée [9 références]

https://hal.inria.fr/hal-01183312
Contributeur : Coordination Episciences Iam <>
Soumis le : mercredi 12 août 2015 - 10:07:13
Dernière modification le : mardi 7 mars 2017 - 15:00:31
Document(s) archivé(s) le : vendredi 13 novembre 2015 - 11:34:52

### Fichier

dmAB0115.pdf
Fichiers éditeurs autorisés sur une archive ouverte

### Identifiants

• HAL Id : hal-01183312, version 1

### Citation

Nick Anzalone, John Baldwin, Ilya Bronshtein, Kyle Petersen. A Reciprocity Theorem for Monomer-Dimer Coverings. Michel Morvan and Éric Rémila. Discrete Models for Complex Systems, DMCS'03, 2003, Lyon, France. Discrete Mathematics and Theoretical Computer Science, DMTCS Proceedings vol. AB, Discrete Models for Complex Systems (DMCS'03), pp.179-194, 2003, DMTCS Proceedings. 〈hal-01183312〉

### Métriques

Consultations de la notice

## 99

Téléchargements de fichiers