Abstract
A polyiamond is an edge-connected set of cells on the triangular lattice. In this paper we provide an improved lower bound on the asymptotic growth constant of polyiamonds, proving that it is at least 2.8424. The proof of the new bound is based on a concatenation argument and on elementary calculus. We also suggest a nontrivial extension of this method for improving the bound further. However, the proposed extension is based on an unproven (yet very reasonable) assumption.
Work on this paper by all authors has been supported in part by ISF Grant 575/15.
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Notes
- 1.
Note that in this reference the lattice is called “honeycomb” (hexagonal) and the terms should be doubled. The reason for this is that the authors actually count clusters of vertices on the hexagonal lattice, whose connectivity is the same as that of cells on the triangular lattice, with no distinction between the two possible orientations of the latter cells. This is why polyiamonds are often regarded in the literature as site animals on the hexagonal lattice, and polyhexes (cell animals on the hexagonal lattice) are regarded as site animals on the triangular lattice, which sometimes causes confusion.
- 2.
This easy upper bound, based on an idea of Eden [7] was described by Lunnon [15, p. 98]. Every polyiamond P can be built according to a set of \(n{-}1\) “instructions” taken from a superset of size \(2(n-1)\). Each instruction tells us how to choose a lattice cell c, neighboring a cell already in P, and add c to P. (Some of these instruction sets are illegal, and some other sets produce the same polyiamonds, but this only helps.) Hence, \(\lambda _T \le \lim _{n \rightarrow \infty } \left( {\begin{array}{c}2(n-1)\\ n-1\end{array}}\right) ^{1/n} = 4\).
- 3.
We wonder why Rands and Welsh did not use \(T(22) = 1,456,891,547\) (which was also available in their reference [19]) to show that \(\lambda _T \ge (T(22)/10)^{1/22} \approx 2.3500\).
- 4.
\(T(75)=15,936,363,137,225,733,301,433,441,827,683,823\).
- 5.
In fact, it is widely believed (but not proven) that the constant \(\theta \) is common to all lattices in the same dimension. In particular, there is evidence that \(\theta = 1\) for all lattices in two dimensions.
- 6.
Madras [16, Proposition 4.2] proved “almost monotonicity” for all lattices, namely, that \(L(n+2)/L(n) \ge (L(n+1)/L(n))^2 - \varGamma _\mathcal{L}/n\) for all sufficiently large values of n, where \(\varGamma _\mathcal{L}\) is a constant which depends on \(\mathcal{L}\). Note that if \(\varGamma _\mathcal{L}= 0\), then we have \(L(n+2)/L(n+1) \ge L(n+1)/L(n)\), i.e., that the ratio sequence of \(\mathcal{L}\) is monotone increasing.
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Barequet, G., Shalah, M., Zheng, Y. (2017). An Improved Lower Bound on the Growth Constant of Polyiamonds. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_5
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