New enumerative results on two-dimensional directed animals

Abstract

We list several open problems concerning the enumeration of directed animals on two-dimensional lattices. We show that most of these problems are special cases of two central problems: calculating the position generating function and the perimeter and area generating function for square lattice animals. We propose a possible direction for solving these two problems: we extend Dhar's correspondence between hard particle gas models and enumeration of animals according to the area, and show that each of the main two generating functions is, essentially, the density of a one-dimensional gas model given by the stationary distribution of a probabilistic transition. We are able to compute the density of certain stationary distributions. We thus obtain new bivariate generating functions for directed animals on the square and triangular lattices. We derive from these results the generating functions for animals on the decorated square and triangular lattices, as well as the average number of loops in directed animals.