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

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

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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