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On Column-Convex and Convex Carlitz Polyominoes

Abstract

In this paper, we introduce and study Carlitz polyominoes. In particular, we show that, as n grows to infinity, asymptotically the number of

  1. (1)

    column-convex Carlitz polyominoes with perimeter 2n is

    $$\begin{aligned} \frac{9\sqrt{2}(14+3\sqrt{3})}{2704\sqrt{\pi n^3}}4^n. \end{aligned}$$
  2. (2)

    convex Carlitz polyominoes with perimeter 2n is

    $$\begin{aligned} \frac{n+1}{10}\left( \frac{3+\sqrt{5}}{2}\right) ^{n-2}. \end{aligned}$$

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References

  1. 1.

    Golomb, S.W.: Checker boards and polyominoes. Am. Math. Monthly 61, 675–682 (1954)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Hakim, V., Nadal, J.P.: Exact results for 2D directed animals on a strip of finite width. J. Phys. A Math. Gen. 16(7), 213–218 (1983)

    Article  Google Scholar 

  3. 3.

    Privman, V., Svrakic, N.M.: Difference equations in statistical mechanics. I. Cluster statistics models. J. Stat. Phys. 51:5–6, 1091–1110 (1988)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Privman, V., Svrakic, N.M.: Directed models of polymers, interfaces, and clusters: scaling and finite-size properties. Springer, Berlin (1989)

    Google Scholar 

  5. 5.

    Viennot, G.: Problémes combinatoires posés par la physique statistique. Astérisque 1(21–122), 225–246 (1985)

    MATH  Google Scholar 

  6. 6.

    Temperley, H.N.V.: Combinatorial problems suggested by the statistical mechanics of domains and of rubber-like molecules. Phys. Rev. 103, 1–16 (1956)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Beauquier, D., Nivat, M., Remila, É., Robson, M.: Tiling figures of the plane with two bars. Comput. Geometry. Theory Appl. 5(1), 1–25 (1995)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Berger, R.: The undecidability of the domino problem. Memoirs Am. Math. Soc. 66, 72 (1966)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Grünbaum, B., Shephard, G.C.: Tilings and patterns. W.H. Freeman and Company, New York (1989)

    MATH  Google Scholar 

  10. 10.

    Klarner, D.A.: My life among the polyominoes. Nieuw Archief voor Wiskunde. Derde Serie 29(2), 156–177 (1981)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Klarner, D.A.: Some results concerning polyominoes. Fib. Quart. 3, 9–20 (1965)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Klarner, D.A.: Packing a rectangle with congruent \(n\)-ominoes. J. Combin. Theory 7, 107–115 (1969)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Barcucci, E., Frosini, A., Rinaldi, S.: Direct-convex polyominoes: ECO method and bijective results. Proc. Formal Power Ser. Algeb. Combin. Melbourne (2002)

  14. 14.

    Conway, A.: Enumerating \(2D\) percolation series by the finite-lattice method: theory. J. Phys. A 28(2), 335–349 (1995)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Delest, M., Viennot, X.G.: Algebraic languages and polyominoes enumeration. Theoret. Comput. Sci. 34, 169–206 (1984)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Guttmann, A.J.: Polygons, Polyominoes and Polycubes. Springer, Netherlands (2009)

    Book  Google Scholar 

  17. 17.

    Viennot, X.G.: A survey of polyominoes enumeration. In: Proceedings of the 4th FPSAC Publications du LACIM, Institut Mittag-Leffler 11, 399–420 (1992)

  18. 18.

    A. Del Lungo, M. Mirolli, R. Pinzani, S. Rinaldi: A bijection for directed-convex polyominoes. In: Proceedings of the DM-CCG 2001, Discrete Mathematics and Theoretical Computer Science AA, pp. 133–144 (2001)

  19. 19.

    Jensen, I.: Enumerations of lattice animals and trees. J. Stat. Phys. 102(3–4), 865–881 (2001)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Jensen, I., Guttmann, A.J.: Statistics of lattice animals (polyominoes) and polygons. J. Phys. A 33(29), 257–263 (2000)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Mansour, T., Rastegar, R.: Convex polyominoes revisited: enumeration of outer site perimeter, interior vertices, and boundary vertices of certain degrees. J. Diff. Eq. Appl. 26(7), 1013–1041 (2021)

    MathSciNet  Article  Google Scholar 

  22. 22.

    S. Feretić, D. Svrtan: On the number of column-convex polyominoes with given perimeter and number of columns. In: Proceedings of the 5th FPSAC, Firenze pp. 201–214 (1993)

  23. 23.

    Feretić, S.: A perimeter enumeration of column-convex polyominoes. Disc. Math. Theoret. Comput. Sci. 9, 57–84 (2007)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Boussicault, A., Rinaldi, S., Socci, S.: The number of directed \(k\)-convex polyominoes. Disc. Math. 343:3, #111731 (2020)

    MathSciNet  Article  Google Scholar 

  25. 25.

    T. Mansour, A. Sh. Shabani: Smooth Column Convex Polyominoes, Submitted

  26. 26.

    Banderier, C., Bousquet-Mélou, M., Denise, A., Flajolet, P., Gardy, D., Gouyou-Beauchamps, D.: Generating functions for generating trees. Disc. Math. 246(1–3), 29–55 (2000)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  28. 28.

    N. Cakić, T. Mansour, G. Yıldırım: A decomposition of column-convex polyominoes and two vertex statistics, preprint

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Correspondence to Toufik Mansour.

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Mansour, T., Rastegar, R. & Shabani, A.S. On Column-Convex and Convex Carlitz Polyominoes. Math.Comput.Sci. 15, 889–898 (2021). https://doi.org/10.1007/s11786-021-00518-z

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Keyword

  • Carlitz polyominoes

Mathematics Subject Classification

  • 05B50
  • 05A16