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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. |

URI: | http://bura.brunel.ac.uk/handle/2438/589 |

ISSN: | 1077-8926 |

Appears in Collections: | Mathematical Science Computer Science |

Files in This Item:

File | Description | Size | Format | |
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forests.pdf | 226.33 kB | Adobe PDF | View/Open |

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