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Title:  Improved bounds for the number of forests and acyclic orientations in the square lattice 
Authors:  Calkin, N Merino, C Noble, S D Noy, M 
Keywords:  Forests Acyclic orientations Square lattice Tutte polynomial Transfer matrices 
Publication Date:  2003 
Publisher:  Electronic Journal of Combinatorics 
Citation:  Electronic Journal of Combinatorics 10(1): R4, Jan 2003 
Abstract:  In a recent paper Merino and Welsh (1999) studied several counting problems on the square lattice $L_n$. 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 $L_n$: $3.209912 \leq \lim_{n\rightarrow\infty} f(n)^{1/n^2} \leq 3.84161$ and $22/7 \leq \lim_{n\rightarrow\infty} \alpha(n) \leq 3.70925$.
In this paper we improve these bounds as follows: $3.64497 \leq \lim_{n\rightarrow\infty} f(n)^{1/n^2} \leq 3.74101$ and $3.41358 \leq \lim_{n\rightarrow\infty} \alpha(n) \leq 3.55449$. We obtain this by developing a method for computing the Tutte polynomial of the square lattice and other related graphs based on transfer matrices. 
URI:  http://bura.brunel.ac.uk/handle/2438/589 
ISSN:  10778926 
Appears in Collections:  Mathematics Information Systems and Computing

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