Please use this identifier to cite or link to this item:
http://bura.brunel.ac.uk/handle/2438/589
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. |
URI: | http://bura.brunel.ac.uk/handle/2438/589 |
ISSN: | 1077-8926 |
Appears in Collections: | Computer Science Mathematical Sciences |
Files in This Item:
File | Description | Size | Format | |
---|---|---|---|---|
forests.pdf | 226.33 kB | Adobe PDF | View/Open |
Items in BURA are protected by copyright, with all rights reserved, unless otherwise indicated.