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

Abstract

In a recent paper Merino and Welsh (1999) studied several counting problems on the square lattice Ln. There the authors gave the following bounds for the asymptotics of f(n), the number of forests of Ln, and α(n), the number of acyclic orientations of Ln: 3.209912 ≤ limn→∞ f(n)1/n2 ≤ 3.84161 and 22/7 ≤ limn→∞ α(n)1/n2 ≤ 3.70925. In this paper we improve these bounds as follows: 3.64497 ≤ limn n→∞ f(n)1/n2 ≤ 3.74101 and 3.41358 ≤ limn→∞ α(n)1/n2 ≤ 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.