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Enumeration of three-dimensional convex polygons

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Abstract

A self-avoiding polygon (SAP) on a graph is an elementary cycle. Counting SAPs on the hypercubic lattice ℤd withd≥2, is a well-known unsolved problem, which is studied both for its combinatorial and probabilistic interest and its connections with statistical mechanics. Of course, polygons on ℤd are defined up to a translation, and the relevant statistic is their perimeter.

A SAP on ℤd is said to beconvex if its perimeter is “minimal”, that is, is exactly twice the sum of the side lengths of the smallest hyper-rectangle containing it. In 1984, Delest and Viennot enumerated convex SAPs on the square lattice [6], but no result was available in a higher dimension.

We present an elementar approach to enumerate convex SAPs in any dimension. We first obtain a new proof of Delest and Viennot's result, which explains combinatorially the form of the generating function. We then compute the generating function for convex SAPs on the cubic lattice. In a dimension larger than 3, the details of the calculations become very cumbersome. However, our method suggests that the generating function for convex SAPs on ℤd is always a quotient ofdifferentiably finite power series.

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Authors and Affiliations

  1. LaBRI, Université Bordeaux 1, 351 cours de la Libération, 33405, Talence Cedex, France

    Mireille Bousquet-Mélou

  2. Department of Mathematics, The University of Melbourne, Parkville, 3052, Victoria, Australia

    Anthony J. Guttmann

Authors
  1. Mireille Bousquet-Mélou
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  2. Anthony J. Guttmann
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Bousquet-Mélou, M., Guttmann, A.J. Enumeration of three-dimensional convex polygons. Annals of Combinatorics 1, 27–53 (1997). https://doi.org/10.1007/BF02558462

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