Annals of Combinatorics 1 (1997) 27-53


Enumeration of Three-dimensional Convex Polygons

Mireille Bousquet-Mélou and Anthony J. Guttmann

LaBRI, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France
bousquet@labri.u-bordeaux.fr

Department of Mathematics, The University of Melbourne, Parkville, Victoria 3052, Australia
tonyg@maths.mu.oz.au

Received July 12, 1996

AMS Subject Classification: 05A15

Abstract. A self-avoiding polygon (SAP) on a graph is an elementary cycle. Counting SAPs on the hypercubic lattice Zd, with d ≥ 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 be convex 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 elementary 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 of differentiably finite power series.

Keywords: self-avoiding polygons, convexity, differentiably finite series


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