On the number of hexagonal polyominoes. (English) Zbl 1053.05028
A combination of the refined finite lattice method and transfer allows a radical increase in the computer enumeration of polyominoes on the hexagonal lattice (equivalently, site clusters on the triangular lattice), \(p_n\), with \(n\) hexagons.
In this paper the authors obtain \(p_n\) for \(n\leq 35\). They prove \(p_n= \iota^{n+o(n)}\), obtain the bounds \(4.8049\leq\iota\leq 5.9047\), and estimate that \(5.1831478\,(17)\).
Finally, they provide compelling numerical evidence that the generating function \(\sum\) \(p_nz^n\approx A(z)\log(1-\iota z)\), for \(z\to (1/\iota)^-\) with \(A(z)\) holomorphic in a cut plane, estimate \(A(1/\iota)\) and predict the subleading asymptotic behavior, identifying a nonanalytic correction-to-scaling term with exponent \(\Delta= 3/2\). On the basis of universality and earlier numerical work, the authors argue that the mean square radius of gyration \(\langle R^2_g\rangle_n\) of polyominoes of size \(n\) grows as \(n^{2v}\), with \(v= 0.64115(5)\).
In this paper the authors obtain \(p_n\) for \(n\leq 35\). They prove \(p_n= \iota^{n+o(n)}\), obtain the bounds \(4.8049\leq\iota\leq 5.9047\), and estimate that \(5.1831478\,(17)\).
Finally, they provide compelling numerical evidence that the generating function \(\sum\) \(p_nz^n\approx A(z)\log(1-\iota z)\), for \(z\to (1/\iota)^-\) with \(A(z)\) holomorphic in a cut plane, estimate \(A(1/\iota)\) and predict the subleading asymptotic behavior, identifying a nonanalytic correction-to-scaling term with exponent \(\Delta= 3/2\). On the basis of universality and earlier numerical work, the authors argue that the mean square radius of gyration \(\langle R^2_g\rangle_n\) of polyominoes of size \(n\) grows as \(n^{2v}\), with \(v= 0.64115(5)\).
Reviewer: Ratnakaram Nava Mohan (Eluru)
MSC:
| 05B50 | Polyominoes |
| 05C30 | Enumeration in graph theory |
| 68R10 | Graph theory (including graph drawing) in computer science |
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