Exact results for the generating function of directed column-convex animals on the square lattice

Abstract

The mathematical properties of the generating function S(x) for the number sn of directed column-convex lattice animals on a square lattice with a given directed-site perimeter n are investigated. In particular, it is shown that S(x) can be expressed exactly in terms of algebraic hypergeometric functions. A detailed investigation of the asymptotic behaviour of sn as n to infinity is carried out by applying the Darboux method to the hypergeometric formula for S(x). It is also demonstrated that sn satisfies a four-term recurrence relation. Finally, it is noted that the techniques used to analyse the lattice animal generating function S(x) can be applied to any other generating function which satisfies a cubic algebraic equation.