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    Improved bounds for the number of forests and acyclic orientations in the square lattice

    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.

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    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: School of Business, Economics & Informatics > Economics, Mathematics and Statistics
    Depositing User: Sarah Hall
    Date Deposited: 20 Jul 2020 12:50
    Last Modified: 20 Jul 2020 12:50
    URI: https://eprints.bbk.ac.uk/id/eprint/32605

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