Abstract
A “lattice animal” is an edge-connected set of cells on a lattice. In this paper we consider the Tetrakis lattice, and provide the first lower bound on \(\lambda _\tau \), the growth constant of polyaboloes (animals on this lattice), proving that \(\lambda _\tau \ge 2.4345\). The proof of the bound is based on a concatenation argument and on calculus manipulations. If we also rely on an unproven assumption, which is, however, supported by empirical data, we obtain the conditional slightly-better lower bound 2.4635.
Work on this paper by the first author has been supported in part by ISF Grant 575/15 and by BSF Grant 2017684. Work by the second author has been supported in part by Grant DST-IFA-14-ENG-75 and new faculty Seed Grant NPN5R.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Originally, the algorithm was proposed for counting polyominoes (site animals on the square lattice). However, as was already noted elsewhere (see, e.g., [2]), this algorithm can be adapted to any lattice, once it is formulated as an algorithm for counting connected subgraphs of a given graph, that contain one marked vertex in the graph. The reader is referred to the reference cited above for further details.
References
The On-Line Encyclopedia of Integer Sequences. http://oeis.org
Aleksandrowicz, G., Barequet, G.: Counting \(d\)-dimensional polycubes and nonrectangular planar polyominoes. Int. J. Comput. Geom. Appl. 19, 215–229 (2009)
Barequet, G., Rote, G., Shalah, M.: An improved upper bound on the growth constant of polyiamonds (2019)
Barequet, G., Shalah, M., Zheng, Y.: An improved lower bound on the growth constant of polyiamonds. J. Comb. Optim. 37, 424–438 (2019)
Barequet, G., Rote, G., Shalah, M.: \(\lambda >4\): an improved lower bound on the growth constant of polyominoes. Commun. ACM 59, 88–95 (2016)
Broadbent, S.R., Hammersley, J.M.: Percolation processes: I. Crystals and mazes. Proc. Cambridge Philos. Soc. 53, 629–641 (1957)
Conway, J.H., Burgiel, H., Goodman-Strass, C.: The Symmetries of Things. A.K. Peters/CRC Press, New York (2008)
Eden, M.: A two-dimensional growth process. In: Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Berkeley CA, vol. IV, pp. 223–239 (1961)
Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W.H. Freeman, New York (1987)
Harary, F.: Unsolved problems in the enumeration of graphs. Math. Inst. Hung. Acad. Sci. 5, 1–20 (1960)
Klarner, D.A.: Cell growth problems. Can. J. Math. 19, 851–863 (1967)
Klarner, D.A., Rivest, R.L.: A procedure for improving the upper bound for the number of \(n\)-ominoes. Can. J. Math. 25, 585–602 (1973)
Madras, N.: A pattern theorem for lattice clusters. Ann. Comb. 3, 357–384 (1999)
Redelmeier, D.H.: Counting polyominoes: yet another attack. Discrete Math. 36, 191–203 (1981)
Temperley, H.N.V.: Combinatorial problems suggested by the statistical mechanics of domains and of rubber-like molecules. Phys. Rev. 2(103), 1–16 (1956)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendix: Computing Elements of \((\tau (n))\)
Appendix: Computing Elements of \((\tau (n))\)
We have implemented Redelmeier’s algorithm [14] for counting polyominoes, and adapted it to the Tetrakis lattice.Footnote 1 The algorithm was implemented in C on a server with four 2.20 GHz Intel Xeon processors and 512 GB of RAM. The software consisted of about 200 lines of code.
Assume, for simplicity, that we wanted to count polyaboloes up to size n, where n is divisible by 4. Then, we created the graph, dual of the portion of the Tetrakis lattice, that consists of n / 2 columns, each of height \(2n+4\). Cells (triangles) of this portion of our lattice were numbered as is shown in Fig. 5(a). In fact, the cell-adjacency graph was identical to the one shown in Fig. 5(b), where a thick edge means that the two cells sharing this edge were not considered as neighbors. In order to count polyaboloes of Types 1, 2, 3, or 4, we fixed their smallest triangle at cell number \(n+1\), \(n+2\), \(n+3\), or n, respectively. This ensured that animals of size n would never spill over the allocated portion of the Tetrakis lattice.
In fact, we needed to count only polyaboloes of two out of the four types, as is implied by Lemma 1. Thus, we ran our program for counting polyaboloes of Types 1 and 2, and computed counts of polyaboloes of Types 3 and 4 by applying the lemma. Then, we summed up the results to finally obtain \(\tau (n) = \sum _{i=1}^4 \tau _i(n)\). The running times of our program were 27.5 and 26.25 days, for computing \(\tau _1(n)\) and \(\tau _2(n)\) for \(1 \le n \le 36\), respectively, for a total of 53.75 days for computing \(\tau (n)\) for this range of n.
Table 2 provides the split of \(\tau (36)\) into all 16 subtypes. Table 3 shows the total counts of polyaboloes, produced by our program, extending significantly the previously-published counts [1, Sequence A197467]. Figure 6 plots the known values of \(\root n \of {\tau (n)}\) and \(\tau (n)/\tau (n-1)\) for \(2 \le n \le 36\), demonstrating the convergence of the two sequences. Figure 7 plots the 36 known values of the sequences \((x_i(n))\) (\(i = 1,\dots ,4\)), showing the tendencies of these sequences.
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Barequet, G., De, M. (2019). A Lower Bound on the Growth Constant of Polyaboloes on the Tetrakis Lattice. In: Du, DZ., Duan, Z., Tian, C. (eds) Computing and Combinatorics. COCOON 2019. Lecture Notes in Computer Science(), vol 11653. Springer, Cham. https://doi.org/10.1007/978-3-030-26176-4_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-26176-4_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-26175-7
Online ISBN: 978-3-030-26176-4
eBook Packages: Computer ScienceComputer Science (R0)