Calkin, N. and Merino, C. and Noble, Steven and Noy, M. (2003) Improved bounds for the number of forests and acyclic orientations in the square lattice. The Electronic Journal of Combinatorics 10 , ISSN 1077-8926.
Abstract
In a recent paper Merino and Welsh (1999) studied 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 α ( n ) , the number of acyclic orientations of L n : 3.209912 ≤ lim n → ∞ f ( n ) 1 / n 2 ≤ 3.84161 and 22 / 7 ≤ lim n → ∞ α ( n ) 1 / n 2 ≤ 3.70925. In this paper we improve these bounds as follows: 3.64497 ≤ lim n → ∞ f ( n ) 1 / n 2 ≤ 3.74101 and 3.41358 ≤ lim n → ∞ α ( n ) 1 / n 2 ≤ 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.
Metadata
| Item Type: | Article |
|---|---|
| Additional Information: | Article number: R4 |
| School: | Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences |
| Depositing User: | Sarah Hall |
| Date Deposited: | 20 Jul 2020 12:50 |
| Last Modified: | 09 Aug 2023 12:48 |
| URI: | https://eprints.bbk.ac.uk/id/eprint/32605 |
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