On the number of hexagonal polyominoes

Abstract

A combination of the refined finite lattice method and transfer matrices allows a radical increase in the computer enumeration of polyominoes on the hexagonal lattice (equivalently, site clusters on the triangular lattice), pn with n hexagons. We obtain pn for n ≤ 35. We prove that pn=τn+o(n), obtain the bounds 4.8049 ≤ τ ≤ 5.9047, and estimate that τ=5.1831478(17). Finally, we provide compelling numerical evidence that the generating function ∑p nzn ≈ A(z)log(1-τz), for z → (1/τ) - with A(z) holomorphic in a cut plane, estimate A(1/τ) and predict the sub-leading asymptotic behaviour, identifying a non-analytic correction-to-scaling term with exponent Δ=3/2. On the basis of universality and ..