Lattice animals and heaps of dimers

The general quest ofthis paper is the search f or new classes ofsquare lattice animals that are both large and exactly enumerable. The starting point is a bijection between a subclass ofanimals, called directed animals, and certain heaps of dimers, called pyramids, which was described by Viennot more than 10 years ago. The generating function for directed animals had been known since 1982, but Viennot’s bijection suggested a new approach that greatly simpli7ed its derivation. We de7ne here two natural classes ofheaps that are supersets ofpyramids and are in bijection with certain classes ofanimals, and we enumerate them exactly. The 7rst class has an algebraic generating function and growth constant 3:5 (meaning that the number of n-celled animals grows like 3:5 n ), while the other has a transcendental non-holonomic generating function and growth constant 3:58 ::: :The generating function for directed animals is algebraic, and has growth constant 3. Hence both these new classes are exponentially larger. We obtain similar results for triangular lattice animals. c 2002 Elsevier Science B.V. All rights reserved.

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