E. Barcucci et al., THE AVERAGE HEIGHT OF DIRECTED COLUMN-CONVEX POLYOMINOES HAVING SQUARE, HEXAGONAL AND TRIANGULAR CELLS, Mathematical and computer modelling, 26(8-10), 1997, pp. 27-36
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.