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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
Issue 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.
ISSN: 1077-8926
Appears in Collections:Mathematical Science
Computer Science

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