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On the number of hexagonal polyominoes

✍ Scribed by Markus Vöge; Anthony J. Guttmann

Publisher
Elsevier Science
Year
2003
Tongue
English
Weight
570 KB
Volume
307
Category
Article
ISSN
0304-3975

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✦ Synopsis

A combination of the reÿned ÿnite 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 6 35. We prove that pn = n+o(n) , obtain the bounds 4:8049 6 6 5:9047, and estimate that =5:1831478(17). Finally, we provide compelling numerical evidence that the generating function pnz n ≈ A(z)log(1-z), for z → (1= ) -with A(z) holomorphic in a cut plane, estimate A(1= ) and predict the sub-leading asymptotic behaviour, identifying a non-analytic correction-to-scaling term with exponent =3=2. On the basis of universality and previous numerical work we argue that the mean-square radius of gyration R 2 g n of polyominoes of size n grows as n 2 , with = 0:64115(5).

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