Annals of Combinatorics 3 (1999) 223-249


Inversion Relations, Reciprocity and Polyominoes

M. Bousquet-Mélou1, A.J. Guttmann2,W.P. Orrick2, and A. Rechnitzer2

1Laboratoire Bordelais de Recherche en Informatique, Université Bordeaux I, 351 cours de la Libération, 33405 Talence Cedex, France
 bousquet@labri.u-bordeaux.fr

2Department of Mathematics and Statistics, The University of Melbourne, Parkville, Victoria 3052, Australia
 tonyg@maths.mu.oz.au, worrick@mailhost.maths.mu.oz.au

Received June 30, 1999

AMS Subject Classification: 05A15, 05B50, 82B20, 82B23

Abstract. We derive self-reciprocity properties for a number of polyomino generating functions, including several families of column-convex polygons, three-choice polygons, and staircase polygons with a staircase hole. In so doing, we establish a connection between the reciprocity results known to combinatorialists and the inversion relations used by physicists to solve models in statistical mechanics. For several classes of convex polygons, the inversion (reciprocity) relation, augmented by certain symmetry and analyticity properties, completely determines the anisotropic perimeter generating function.

Keywords: nversion relations, combinatorial reciprocity theorems, polyominoes, self-avoiding polygons, convex polygons, statistical mechanics


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