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