The average height of directed column-convex polyominoes having square, hexagonal and triangular cells

Abstract

There is a well-known correspondence between animals on the square lattice and polyominoes having square cells. Since the animals have also been defined on triangular and hexagonal lattices, in this paper, we are going to examine their corresponding polyominoes. We examine the enumeration of directed column-convex square, hexagonal and triangular polyominoes according to their area, number of columns and height. By means of a recursive description of these polyominoes, we obtain a functional equation verified by their generating function. From the equations obtained, we deduce the average height of directed column-convex polyominoes having a fixed area for each lattice. In each family of polyominoes, the asymptotic average height ξ ∥ of its polyominoes usually defines a critical exponent ν ∥ in the form of ξ ∥ ( n) ≈ n ν ∥ . We find that the critical exponent ν ∥ is equal to 1 for all the three lattices. These results confirm the “universal hypothesis” made by some physicists and, to the authors' knowledge, represent the first exact results regarding the average height of directed polyominoes.