A Reciprocity Theorem for Monomer-Dimer Coverings

We prove that the number of monomer-dimer tilings of an $n\times n$ square grid, with $m<n$ monomers in which no four tiles meet at any point is $m2^m+(m+1)2^{m+1}$, when $m$ and $n$ have the same parity. In addition, we present a new proof of the result that there are $n2^{n-1}$ such tilings with $n$ monomers, which divides the tilings into $n$ classes of size $2^{n-1}$. The sum of these tilings over all monomer counts has the closed form $2^{n-1}(3n-4)+2$ and, curiously, th...

We use computational method to investigate the number of ways to pack dimers on \emph{odd-by-odd} lattices. In this case, there is always a single vacancy in the lattices. We show that the dimer configuration numbers on $(2k+1) \times (2k+1)$ \emph{odd} square lattices have some remarkable number-theoretical properties in parallel to those of close-packed dimers on $2k \times 2k$ \emph{even} square lattices, for which exact solution exists. Furthermore, we demonstrate that th...

We present a innovative relationship between ground states of the Ising model and dimer coverings which sheds new light on the Ising Models with highly degenerated ground states and enables one to construct such models. Thanks to this relationship we also find the generating function for dimers as the appropriate limit of the free energy per spin for the Ising model.

We construct a class of lattices in three and higher dimensions for which the number of dimer coverings can be determined exactly using elementary arguments. These lattices are a generalization of the two-dimensional kagome lattice, and the method also works for graphs without translational symmetry. The partition function for dimer coverings on these lattices can be determined also for a class of assignments of different activities to different edges.

In this paper, we explore some generalizations of a counting problem related to tilings in grids of size 2xn, which was originally posed as a question on Mathematics Stack Exchange (Question 3972905). In particular, we consider this problem for the product of two graphs G and P(n), where P(n) is the path graph of n vertices. We give explicit bivariate generating functions for some specific cases.

We study perfect multiple coverings in translation invariant graphs with vertex set $\mathbb{Z}^2$ using an algebraic approach. In this approach we consider any such covering as a two-dimensional binary configuration which we then express as a two-variate formal power series. Using known results, we conclude that any perfect multiple covering has a non-trivial periodizer, that is, there exists a non-zero polynomial whose formal product with the power series presenting the cov...

Using a relation between the virial expansion coefficients of the pressure and the entropy expansion coefficients in the case of the monomer-dimer model on infinite regular lattices, we have shown that, on hypercubic lattices of any dimension, the virial coefficients are positive through the 20th order. We have observed that all virial coefficients so far known for this system are positive also on infinite regular lattices with different structure. We are thus led to conjectu...

Dimer coverings (or perfect matchings) of a finite graph are classical objects of graph theory appearing in the study of exactly solvable models of statistical mechanics. We introduce more general dimer labelings which form a topological space called the dimer space of the graph. This space turns out to be a cubed complex whose vertices are the dimer coverings. We show that the dimer space is nonpositively curved in the sense of Gromov, so that its universal covering is a CAT...

The dimer (monomer-dimer) model deals with weighted enumeration of perfect matchings (matchings). The monopole-dimer model is a signed variant of the monomer-dimer model whose partition function is a determinant. In 1999, Lu and Wu evaluated the partition function of the dimer model on two-dimensional grids embedded on a M\"obius strip and a Klein bottle. While the partition function of the dimer model has been known for the two-dimensional grids with different boundary condi...

In the last decade there have been many results about special families of graphs whose number of perfect matchings is given by perfect or near perfect powers. In this paper we present an approach that allows proving them in a unified way. We use this approach to prove a conjecture of James Propp stating that the number of tilings of the so-called Aztec dungeon regions is a power (or twice a power) of 13. We also prove a conjecture of Matt Blum stating that the number of perfe...