THE AVERAGE HEIGHT OF DIRECTED COLUMN-CONVEX POLYOMINOES HAVING SQUARE, HEXAGONAL AND TRIANGULAR CELLS

There is a well-known correspondence between animals on the square lat tice and polyominoes having square cells. Since the animals have also been defined on triangular and hexagonal lattices, in this paper, we a re going to examine their corresponding polyominoes. We examine the en umeration of directed column-convex square, hexagonal and triangular p olyominoes according to their area, number of columns and height. By m eans of a recursive description of these polyominoes, we obtain a func tional equation verified by their generating function. From the equati ons obtained, we deduce the average height of directed column-convex p olyominoes having a fixed area for each lattice. In each family of pol yominoes, the asymptotic average height xi(parallel to) of its polyomi noes usually defines a critical exponent nu(parallel to) in the form o f xi(parallel to)(n) approximate to n(nu parallel to). We find that th e critical exponent nu(parallel to) is equal to 1 for all the three la ttices. These results confirm the ''universal hypothesis'' made by som e physicists and, to the authors' knowledge, represent the first exact results regarding the average height of directed polyominoes.