Improved bounds for the number of forests and acyclic orientations in the square lattice (Q1856350)

A recent paper by \textit{C. Merino} and \textit{D. J. A. Welsh} [Ann. Comb. 3, No. 2-4, 417-429 (1999; Zbl 0936.05043)] considered 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 \(L_n\): \[ \begin{aligned} 3.209912 &\leq \lim_{n\to\infty} f(n)^{1/n^2}\leq 3.84161\\ 22/7 &\leq \lim_{n\to\infty} \alpha(n)^{1/n^2}\leq 3.70925.\end{aligned} \] In this paper, the authors improve these bounds as follows: \[ \begin{aligned} 3.64497 &\leq \lim_{n\to\infty} f(n)^{1/n^2}\leq 3.74101\\ 3.41358 &\leq \lim_{n\to\infty} \alpha(n)^{1/n^2}\leq 3.55449.\end{aligned} \] They obtain these improved bounds by developing a method for computing the Tutte polynomial of the square lattice and other related graphs based on transfer matrices.