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{\bf N. Calkin, C. Merino, S. Noble and M. Noy}
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{\bf Improved Bounds for the Number of Forests and Acyclic Orientations in the Square Lattice}
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In a recent paper Merino and Welsh (1999) studied
several counting problems on the square lattice $L_n$. There the
authors gave the following bounds for the asymptotics of $f(n)$, the
number of forests of $L_n$, and $\alpha(n)$, the number of acyclic
orientations of $L_n$: 
$$3.209912 \le \lim_{n\to\infty} f(n)^{1/n^2}
\le 3.84161$$ and $$22/7 \le \lim_{n\to\infty} \alpha(n)^{1/n^2} \le 3.70925.$$

 In this paper we improve these bounds as follows:
$$3.64497 \le \lim_{n\to\infty} f(n)^{1/n^2} \le 3.74101$$ and $$3.41358
\le \lim_{n\to\infty} \alpha(n)^{1/n^2} \le 3.55449.$$
 We obtain this by developing a method for computing the Tutte
 polynomial of the  square lattice and other related graphs based
 on transfer matrices.

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