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An improved lower bound on the growth constant of polyiamonds

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Abstract

A polyiamond is an edge-connected set of cells on the triangular lattice in the plane. 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.

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Notes

  1. Note that in this reference, the lattice is called “honeycomb” (hexagonal), and that the terms provided there 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 triangular 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 (1961) was described by Lunnon (1972, p. 98), and it is justified as follows. 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. \(T(75)=15,936,363,137,225,733,301,433,441,827,683,823\).

  4. The preliminary version of this paper (Barequet et al. 2017) contained an error, claiming that the logical entailment in item (i) of this theorem is bidirectional.

  5. In fact, it is widely believed (but has never been proven either) 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 (1999, Prop. 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 proving that \(\varGamma _{{\mathcal {L}}}= 0\) would imply that \(L(n+2)/L(n+1) \ge L(n+1)/L(n)\), i.e., that the ratio sequence of \({{\mathcal {L}}}\) is monotone increasing.

  7. A proper pattern is a subgraph P of the lattice, described by two parts: one which must be present in P, and the other, possibly empty, which must not be present in P.

  8. In this transformation, each triangle (cell) in \({{\mathcal {T}}}\) is mapped to a vertex (site) in \({{\mathcal {H}}}\), each edge (cell neighborhood) in \({{\mathcal {T}}}\) is mapped to an edge (bond) in \({{\mathcal {H}}}\), and each vertex in \({{\mathcal {T}}}\) is mapped to a hexagon (cell) in \({{\mathcal {H}}}\). Thus, a polyiamond on \({{\mathcal {T}}}\) is mapped bijectively to a cite animal on \({{\mathcal {H}}}\). The distinction between the two types of triangles on \({{\mathcal {T}}}\) is mapped to the distinction between the two types of vertices on \({{\mathcal {H}}}\), made by the relative orientation of the three bonds connected to a cite (namely, the orientation of the “fork”). The lexicographic order of triangles on \({{\mathcal {T}}}\) is mapped to the usual coordinates-based lexicographic order of vertices on \({{\mathcal {H}}}\).

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Acknowledgements

We wish to thank Neal Madras for helpful discussions about the use of his pattern and ratio-limit theorems in the context of polyiamonds. We also thank an anonymous referee for pointing out a flaw in Theorem 4.

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Correspondence to Gill Barequet.

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Work on this paper by all authors has been supported in part by ISF Grant 575/15. A preliminary version of this paper appeared in COCOON’17 (Barequet et al. 2017).

Appendix: convergence of the sequence \(\varvec{(T_1(n+1)/T_1(n))}\)

Appendix: convergence of the sequence \(\varvec{(T_1(n+1)/T_1(n))}\)

In 1999, Madras (1999) proved a Ratio Limit Theorem, showing the existence of the growth constant (limit of the ratio sequence) for animals on any periodic lattice \({{\mathcal {L}}}\), namely, that \(\lim _{n \rightarrow \infty } L(n+1)/L(n)\) (where L(n) is the number of animals on \({{\mathcal {L}}}\)) exists. The same method can be used to prove the existence of the growth constant of the sequences enumerating some restricted family of animals, provided that they fulfill some set of conditions. We show here the application of this method to polyiamonds of Type 1 on the planar triangular lattice.

Theorem 2The sequence\((T_1(n+1)/T_1(n))_{n=1}^{\infty }\)converges.

Fig. 5
figure 5

Modified concatenation rules for Cluster Axiom 3 (Madras 1999) (Color figure online)

Proof

Our goal is to show that the set of polyiamonds of Type 1 fulfills all five Cluster Axioms and one additional condition, listed in the reference cited above.

The first axiom is held by definition: The set of polyiamonds of Type 1 of size n is a finite collection of subgraphs of the triangular lattice, which are invariant under translation.

The second axiom is also held in a trivial manner: If we assign the same weight to all polyiamonds of Type 1, then the difference in weight between “similar” such polyiamonds is obviously bounded.

The third axiom (the existence of \(\lim _{n \rightarrow \infty } (T_1(n))^{1/n}\)) is easily shown by a standard concatenation argument. The fact that the smallest triangle of some polyiamonds is of the “wrong” type, i.e., Type 1, can easily be overcome by modifying the concatenation rules. Refer to Fig. 5 which shows a few cases of concatenating two n-cell polyiamonds (\(n > 1\)) of Type 1. Let \(P_1\) denote the left (lexicographically smaller) polyiamond. Since \(P_1\) is of Type 1, its largest triangle, denoted by a, is of Type 1 by definition. The polyiamond \(P_1\) must also contain a triangle b, of Type 2, located immediately below a. Let \(P_2\) denote the right (lexicographically larger) polyiamond. If c, the smallest triangle of \(P_2\), is of Type 2 (see Fig. 5a.1), then we attach c to the right side of a (Fig. 5a.2) as usual. Consider now the problematic cases, in which the smallest triangle of the right polyiamond \(P_2\) is of Type 1. The polyiamond \(P_2\)may contain a triangle d, of Type 2, immediately above c. If triangle d is present in \(P_2\) (see Fig. 5b.1), then we attach the left side of d to the right side of a (Fig. 5b.2). Although this operation is not a standard concatenation of two polyiamonds, it still yields a valid polyiamond of size 2n. Otherwise, if triangle d is not present in \(P_2\) (as in Fig. 5c.1), then we overlap triangles a and c, and add triangle d to the resulting shape (Fig. 5c.2), which is, again, a valid polyiamond of size 2n. It is easy to verify that the resulting polyiamonds of size 2n in all cases are distinct. Indeed, in different cases, the positions of the \((n{+}1)\)st triangle, relative to the nth triangle (both colored red in Fig. 5a.2, b.2, c.2), are different. The calculation in the concatenation argument now proceeds as usual.

Familiarity of the reader with the terminology and techniques used in Madras’s work is required for the discussion of Axioms 4 and 5.

The fourth axiom is held as well. The axiom requires that any part of any cluster can be changed locally so as to create an occurrence of a translated copy of any so-called proper pattern.Footnote 7 Such a change is guaranteed to exist within the set of all polyiamonds, and, for our purposes, we need to show that it also exists within the set of polyiamonds of Type 1 only. Suppose, then, that we start from a polyiamond of Type 1 and the guaranteed change results in a polyiamond of Type 2. To fit Madras’s terminology, we represent polyiamonds as connected subgraphs on the hexagonal lattice \({{\mathcal {H}}}\), which is the dual of the triangular lattice \({{\mathcal {T}}}\).Footnote 8 Madras describes how to change a part of any cluster (animal) to agree with a given pattern. His method uses a square bounding box, D, of the pattern, translated to the part of the cluster that is about to change. The inside of the box is replaced by the pattern, and the boundary of the box, \(\partial D\), is added to the cluster in order to ensure the connectivity of the resulting animal. In our setting, the potential problem is that this operation may result in a polyiamond of Type 2, manifested on \({{\mathcal {H}}}\) by v, the largest vertex of the new animal, being of the wrong type. This may happen if v is not the largest vertex of the original animal (which is known to be of Type 1), but the largest vertex of \(\partial D\). However, this can be avoided quite easily, either by choosing the origin of coordinates so that the top-right (largest) vertex of any axis-aligned square, whose corners have integer coordinates, is always of the desired Type 1, or by using, if necessary, a slightly-modified box, namely (on \({{\mathcal {H}}}\)) the original box D plus one vertex located immediately above the top-right vertex of D.

The fifth axiom is also held as follows. This axiom requires that for any cluster, there always exist two proper patterns, whose unions of compulsory and forbidden parts are equal, such that if the cluster contains any of these patterns, then replacing in it this pattern by the other: (1) yields a valid (continuous) cluster; (2) reduces (resp., enlarges) the number of occurrences of the former (resp., latter) pattern in the cluster; (3) enlarges the size of the cluster by one. (The fourth assertion refers to the weight of clusters, but it is fulfilled trivially since in our context all clusters have the same weight.) Madras provides a pair of proper patterns which fulfills this axiom, again, with an argument involving a bounding box. As with the fourth axiom, we face the same problem that replacing the contents of a box in a polyiamond of Type 1 may result in a polyiamond of Type 2. However, this can be avoided in the same way described above.

Finally, we trivially have (using, again, a simple concatenation argument), that there exists a constant \(c > 1\), for which we have that \(T_1(n+1) \ge c T_1(n)\) for all \(n \in {\mathbb {N}}\).

This is sufficient for showing that Madras’s Ratio Limit Theorem (Madras 1999, Thm. 2.2, p. 367) can be applied to polyiamonds of Type 1. \(\square \)

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Barequet, G., Shalah, M. & Zheng, Y. An improved lower bound on the growth constant of polyiamonds. J Comb Optim 37, 424–438 (2019). https://doi.org/10.1007/s10878-018-0336-0

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