Polygonal polyominoes on the square lattice

Abstract

We study a proper subset of polyominoes, called polygonal polyominoes, which are defined to be self-avoiding polygons containing any number of holes, each of which is a self-avoiding polygon. The staircase polygon subset, with staircase holes, is also discussed. The internal holes have no common vertices with each other, nor any common vertices with the surrounding polygon. There are no `holes-within-holes'. We use the finite-lattice method to count the number of polygonal polyominoes on the square lattice. Series have been derived for both the perimeter and area generating functions. It is known that while the critical point is unchanged by a finite number of holes, when the number of holes is unrestricted the critical point changes. The area generating function coefficients grow exponentially, with a growth constant greater than that for polygons with a finite number of holes, but less than that of polyominoes. We provide an estimate for this growth constant and prove that it is strictly less than that for polyominoes. Also, we prove that, enumerating by perimeter, the generating function of polygonal polyominoes has zero radius of convergence and furthermore we calculate the dominant asymptotics of its coefficients using rigorous bounds.