Advertisement

Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

Conference paper
  • 212 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12118)

Abstract

A d-dimensional polycube is a face-connected set of cells on \(\mathbb {Z}^d\). Let \(A_d(n)\) denote the number of d-dimensional polycubes (distinct up to translations) with n cubes, and \(\lambda _d\) denote their growth constant \( \lim _{n \rightarrow \infty }\frac{A_d(n{+}1)}{A_d(n)}\). We revisit and extend the method for the best known upper bound on \(A_2(n)\). Our contributions: We (1) prove that \(\lambda _2 \le 4.5252\); (2) prove that \(\lambda _d \le (2d-2)e+o(1)\) for \(d \ge 2\) (already improving significantly the upper bound on \(\lambda _3\) to 9.8073); and (3) implement an iterative process in 3D, improving further the upper bound on \(\lambda _3\) to 9.3835.

Keywords

Klarner’s constant Square lattice Cubical lattice 

References

  1. 1.
    The On-line Encyclopedia of Integer Sequences. http://oeis.org
  2. 2.
    Aleksandrowicz, G., Barequet, G.: Counting \(d\)-dimensional polycubes and nonrectangular planar polyominoes. Int. J. Comput. Geom. Appl. 19, 215–229 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Aleksandrowicz, G., Barequet, G.: Counting polycubes without the dimensionality curse. Discret. Math. 309, 576–583 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barequet, G., Moffie, M., Ribó, A., Rote, G.: Counting polyominoes on twisted cylinders. INTEGERS: Electron. J. Comb. Number Theory 6, 37 (2006)Google Scholar
  5. 5.
    Barequet, G., Rote, G., Shalah, M.: \(\lambda > 4\): an improved lower bound on the growth constant of polyominoes. Commun. ACM 59, 88–95 (2016)CrossRefGoogle Scholar
  6. 6.
    Barequet, R., Barequet, G., Rote, G.: Formulae and growth rates of high-dimensional polycubes. Combinatorica 30, 257–275 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Conway, A.: Enumerating 2D percolation series by the finite-lattice method: theory. J. Phys. A: Math. General 28, 335–349 (1995)Google Scholar
  8. 8.
    Eden, M.: A two-dimensional growth process. In: Neyman, J. (ed.) Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. 4, pp. 223–239 (1961)Google Scholar
  9. 9.
    Gaunt, D., Sykes, M., Ruskin, H.: Percolation processes in \(d\)-dimensions. J. Phys. A: Math. General 9, 1899–1911 (1976)Google Scholar
  10. 10.
    Guttmann, A. (ed.): Polygons, Polyominoes, and Polycubes, vol. 775. Springer, Dordrecht (2009).  https://doi.org/10.1007/978-1-4020-9927-4CrossRefzbMATHGoogle Scholar
  11. 11.
    Jensen, I.: Counting polyominoes: a parallel implementation for cluster computing. In: Sloot, P.M.A., Abramson, D., Bogdanov, A.V., Gorbachev, Y.E., Dongarra, J.J., Zomaya, A.Y. (eds.) ICCS 2003. LNCS, vol. 2659, pp. 203–212. Springer, Heidelberg (2003).  https://doi.org/10.1007/3-540-44863-2_21CrossRefGoogle Scholar
  12. 12.
    Klarner, D.: Cell growth problems. Can. J. Math. 19, 851–863 (1967)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Klarner, D., Rivest, R.: A procedure for improving the upper bound for the number of \(n\)-ominoes. Can. J. Math. 25, 585–602 (1973)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lubensky, T., Isaacson, J.: Statistics of lattice animals and dilute branched polymers. Phys. Rev. A 20, 2130–2146 (1979)CrossRefGoogle Scholar
  15. 15.
    Luther, S., Mertens, S.: Counting lattice animals in high dimensions. J. Stat. Mech.: Theory Exp. 9, 546–565 (2011)Google Scholar
  16. 16.
    Madras, N.: A pattern theorem for lattice clusters. Ann. Comb. 3, 357–384 (1999)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Madras, N., et al.: The free energy of a collapsing branched polymer. J. Phys. A: Math. General 23, 5327–5350 (1990)Google Scholar
  18. 18.
    Mertens, S., Lautenbacher, M.: Counting lattice animals: a parallel attack. J. Stat. Phys. 66, 669–678 (1992)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Rands, B., Welsh, D.: Animals, trees and renewal sequences. IMA J. Appl. Math. 27, 1–17 (1981)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Read, R.: Contributions to the cell growth problem. Can. J. Math. 14, 1–20 (1962)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Redelmeier, D.: Counting polyominoes: yet another attack. Discret. Math. 36, 191–203 (1981)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sykes, M., Glen, M.: Percolation processes in two dimensions: I. low-density series expansions. J. Phys. A: Math. Gen. 9, 87–95 (1976)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Computer ScienceThe Technion—Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of Computer ScienceStanford UniversityStanfordUSA

Personalised recommendations