%bdd.tex: The Number of Monomer-Dimer Tilings of an m by n Rectangle with a Fixed Proportion of Monomers
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{\bf  
The Number of Monomer-Dimer Tilings of an m by n Rectangle with a Fixed Proportion of Monomers
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{\it Manuel KAUERS, Gleb POGUDIN and Doron ZEILBERGER}



{\bf Abstract}: We use symbolic and numeric computations to study the 
asymptotic behavior of the number of monomer-dimer tilings of a strip
of fixed width with a fixed proportion of monomers, and study the ``entropy''
as a function of the propotion, and try to extrapolate the date to the
whole plane (for a strip of width infinity).

{\bf Introduction}

The  problem of enumerating dimer tilings is one of the most celebrated problems in both combinatorics
and statistical physics, that was solved completely by Kasteleyan and Temperly \& Fisher ([K][FT]).
The corresponding problem for the {\it monomer-dimer} problem is wide open, and is believed to be
{\bf intractable}.

Nevetheless, it is possible to say {\it something} about it. 

In this article we will study the following problem.

Fix a rational number $k=a/b$, between $0$ and $1$, and a positive integer $m$, how many ways
can you tile an $m$ by $2nb$ using dimers (dominoes, i.e. $2 \times 1$ and $1 \times 2$ tiles) 
and monomers ($1 \times 1$ tiles), with {\it exactly} $2nbm\cdot k=nma$ dimers?
Let's call that number $T_{k,m}(2nb)$

We are mainly interested in the {\it entropy}
$$
\lambda_{2,m} (k):= \lim\limits_{n \to \infty} \frac{\log T_{k,m}(2nb)}{2mnb}  \quad .
$$


We will compute if for many rational numbers $k$, and for $m \leq 8$, and look at the trend, and
try to estimate the limiting function  $\lambda_{2,\infty}(k)=\lambda_2(k)$.

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{\bf Supporting Maple Package and Output}

All the results in this article were obtained by the use of the Maple packages

$\bullet$ {\tt http://www.math.rutgers.edu/\~{}zeilberg/tokhniot/BDD.txt} \quad ,

whose output files, along with links to diagrams, are available from the {\it front} of this article

{\tt http://www.math.rutgers.edu/\~{}zeilberg/mamarim/mamarimhtml/bdd.html} \quad .


{\bf Acknowledgment}
We wish thank Manuel Kauers for helpful discussions, and for introducing the authors
to each other.




{\bf References}

[FT] M. Fisher and H. Temperley, {\it Dimer Problems in Statistical
Mechanics-an exact result}, Philos. Mag. {\bf 6} (1961), 1061-1063.

[K] P. W. Kasteleyn, {\it The statistics of dimers on a lattice: I. The number of
dimer arrangements in a quadratic lattice}, Physica {\bf 27} (1961), 1209-1225.


[J] Mark Jerrum, {\it Two-dimensional monomer-dimer systems are computationally intractable}, Journal of Statistical Physics, {\bf 48} 121–134.

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\hrule
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Manuel Kauers, Institute of Algebra, Johannes Kepler University, 
Email: {\tt mkauers@kauers.de }   \quad .


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Gleb Pogudin, Institute of Algebra, Johannes Kepler University, 
Email: {\tt pogudin.gleb@gmail.com }   \quad .
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Doron Zeilberger, Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen
Rd., Piscataway, NJ 08854-8019, USA. \hfill\break
Email: {\tt DoronZeil at gmail dot com}   \quad .


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