Exactly Solved Models

  • Mireille Bousquet-Mélou
  • Richard Brak
Part of the Lecture Notes in Physics book series (LNP, volume 775)
This chapter deals with the exact enumeration of certain classes of (self-avoiding) polygons and polyominoes. We restrict our attention to the square lattice. As the interior of a polygon is a polyomino, we often consider polygons as special poly-ominoes. The usual enumeration parameters are the area (the number of cells) and the perimeter (the length of the border). The perimeter is always even, and often refined into the horizontal and vertical perimeters (number of horizontal/vertical steps in the border). Given a class C of polyominoes, the objective is to determine the following complete generating function of C:
$$C\left( {x,y,q} \right) = \sum\limits_{P \in d} {x^{hp\left( P \right)/2_y vp\left( P \right)/2_q a\left( P \right)},} $$
where hp(P), vp(P) and a(P) respectively denote the horizontal perimeter, the vertical perimeter and the area of P. This means that the coefficient c(m,n,k) of x m y n q k in the series C(x,y,q) is the number of polyominoes in the class C having horizontal perimeter 2m, vertical perimeter 2n and area k. Several specializations of C(x,y,q) may be of interest, such as the perimeter generating function C(t,t,1), its anisotropic version C(x,y,1), or the area generating function C(1,1,q). From such exact results, one can usually derive many of the asymptotic properties of the poly-ominoes of C : for instance the asymptotic number of polyominoes of perimeter n, or the (asymptotic) average area of these polyominoes, or even the limiting distribution of this area, as n tends to infinity (see Chapter 11). The techniques that are used to derive asymptotic results from exact ones are often based on complex analysis. A remarkable survey of these techniques is provided by Flajolet and Sedgewick's book [33].


Formal Power Series Convex Polygon Growth Constant Rightmost Column Recursive Construction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. E. Alm and S. Janson. Random self-avoiding walks on one-dimensional lattices. Comm. Statist. Stochastic Models, 6(2):169–212, 1990.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    G. E. Andrews. The theory of partitions. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2.Google Scholar
  3. 3.
    C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy, and D. Gouyou-Beauchamps. Generating functions for generating trees. Discrete Math., 246(1–3):29–55, 2002.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    C. Banderier and P. Flajolet. Basic analytic combinatorics of directed lattice paths. Theoret. Comput. Sci., 281(1–2):37–80, 2002.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    G. Barequet, M. Moffie, A. Ribo, and G. Rote. Counting polyominoes on twisted cylinders. Integers, 6:A22, 37 pp. (electronic), 2006.MathSciNetGoogle Scholar
  6. 6.
    E. A. Bender. Convex n-ominoes. Discrete Math., 8:219–226, 1974.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    J. Bétréma and J.-G. Penaud. Modèles avec particules dures, animaux dirigés et séries en variables partiellement commutatives. ArXiv:math.CO/0106210.Google Scholar
  8. 8.
    M. Bousquet-Mélou. Rapport scientifique d'habilitation. Report 1154-96, LaBRI, Universite Bordeaux 1,
  9. 9.
    M. Bousquet-Mélou. Codage des polyominos convexes et equations pour l'énumération suiv-ant l'aire. Discrete Appl. Math., 48(1):21–43, 1994.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    M. Bousquet-Mélou. A method for the enumeration of various classes of column-convex polygons. Discrete Math., 154(1-3): 1–25, 1996.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    M. Bousquet-Melou. New enumerative results on two-dimensional directed animals. Discrete Math, 180(1–3):73–106, 1998.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    M. Bousquet-Melou. Rational and algebraic series in combinatorial enumeration. In Proceedings of the International Congress of Mathematicians, pages 789–826, Madrid, 2006. European Mathematical Society Publishing House.Google Scholar
  13. 13.
    M. Bousquet-Melou and A. J. Guttmann. Enumeration of three-dimensional convex polygons. Ann. Comb., 1(1):27–53, 1997.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    M. Bousquet-Melou and M. Petkovšek. Linear recurrences with constant coefficients: the multivariate case. Discrete Math., 225(1–3):51–75, 2000.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    M. Bousquet-Mélou and A. Rechnitzer. Lattice animals and heaps of dimers. Discrete Math., 258(1–3):235–274, 2002.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    M. Bousquet-Mélou and X. G. Viennot. Empilements de segments et q-énumération de polyominos convexes dirigés. J. Combin. Theory Ser A, 60(2): 196–224, 1992.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    R. Brak and A. J. Guttmann. Exact solution of the staircase and row-convex polygon perimeter and area generating function. J. PhysA: Math. Gen, 23(20):4581–4588, 1990.MATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    R. Brak, A. L. Owczarek, and T. Prellberg. Exact scaling behavior of partially convex vesicles. J. Stat. Phys., 76(5/6): 1101–1128, 1994.MATHCrossRefADSGoogle Scholar
  19. 19.
    A. de Mier and M. Noy. A solution to the tennis ball problem. Theoret. Comput. Sci., 346(2– 3):254–264, 2005.MATHMathSciNetGoogle Scholar
  20. 20.
    A. Del Lungo, M. Mirolli, R. Pinzani, and S. Rinaldi. A bijection for directed-convex polyominoes. In Discrete models: Combinatorics, Computation, and Geometry (Paris, 2001), Discrete Math. Theor Comput. Sci. Proc, pages 133–144 (electronic). Maison Inform. Math. Discr, Paris, 2001.Google Scholar
  21. 21.
    M. Delest and S. Dulucq. Enumeration of directed column-convex animals with given perimeter and area. CroaticaChemicaActa, 66(1):59–80, 1993.Google Scholar
  22. 22.
    M.-P. Delest. Generating functions for column-convex polyominoes. J. Combin. Theory Ser. A, 48(1): 12–31, 1988.CrossRefMathSciNetGoogle Scholar
  23. 23.
    M.-P. Delest and G. Viennot. Algebraic languages and polyominoes enumeration. Theoret. Comput. Sci., 34(1–2): 169–206, 1984.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    D. Dhar. Equivalence of the two-dimensional directed-site animal problem to Baxter's hard square lattice gas model. Phys. Rev. Lett, 49:959–962, 1982.CrossRefADSMathSciNetGoogle Scholar
  25. 25.
    D. Dhar. Exact solution of a directed-site animals-enumeration problem in three dimensions. Phys. Rev. Lett, 51(10):853–856, 1983.CrossRefADSMathSciNetGoogle Scholar
  26. 26.
    E. Duchi and S. Rinaldi. An object grammar for column-convex polyominoes. Ann. Comb., 8(1):27–36, 2004.MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    I. G. Enting and A. J. Guttmann. On the area of square lattice polygons. J. Statist. Phys., 58(3–4):475–484, 1990.MATHCrossRefADSMathSciNetGoogle Scholar
  28. 28.
    S. Feretić. The column-convex polyominoes perimeter generating function for everybody. Croatica ChemicaActa, 69(3):741–756, 1996.Google Scholar
  29. 29.
    S. Feretić. A new way of counting the column-convex polyominoes by perimeter. Discrete Math, 180(1–3): 173–184, 1998.MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    S. Feretic. An alternative method for q-counting directed column-convex polyominoes. Discrete Math, 210(1–3):55–70, 2000.MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    S. Feretić. A q-enumeration of convex polyominoes by the festoon approach. Theoret. Com-put. Sci., 319(1–3):333–356, 2004.MATHCrossRefGoogle Scholar
  32. 32.
    S. Feretić and D. Svrtan. On the number of column-convex polyominoes with given perimeter and number of columns. In Barlotti, Delest, and Pinzani, editors, Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, Italy), pages 201–214, 1993.Google Scholar
  33. 33.
    P. Flajolet and R. Sedgewick. Analytic Combinatorics. Preliminary version available at
  34. 34.
    G. Gasper and M. Rahman. Basic hypergeometric series, volume 35 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1990.Google Scholar
  35. 35.
    A. J. Guttmann and T. Prellberg. Staircase polygons, elliptic integrals, Heun functions and lattice Green functions. Phys. Rev. E, 47:R2233–R2236, 1993.CrossRefADSGoogle Scholar
  36. 36.
    I. Jensen and A. J. Guttmann. Self-avoiding polygons on the square lattice. J. Phys. A, 32(26):4867–4876, 1999.MATHCrossRefMathSciNetADSGoogle Scholar
  37. 37.
    I. Jensen and A. J. Guttmann. Statistics of lattice animals (polyominoes) and polygons. J. Phys. A, 33(29):L257–L263, 2000.MATHCrossRefMathSciNetADSGoogle Scholar
  38. 38.
    D. A. Klarner. Some results concerning polyominoes. Fibonacci Quart., 3:9–20, 1965.MATHMathSciNetGoogle Scholar
  39. 39.
    D. A. Klarner. Cell growth problems. Canad. J. Math., 19:851–863, 1967.MATHMathSciNetCrossRefGoogle Scholar
  40. 40.
    D. A. Klarner and R. L. Rivest. Asymptotic bounds for the number of convex n-ominoes. Discrete Math., 8:31–40, 1974.MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Y. Le Borgne and J.-F. Marckert. Directed animals and gas models revisited. Electron. J. Combin., R71, 2007.Google Scholar
  42. 42.
    N. Madras and G. Slade. The self-avoiding walk. Probability and its Applications. Birkhaüser Boston Inc., Boston, MA, 1993.Google Scholar
  43. 43.
    G. Pólya. On the number of certain lattice polygons. J. Combinatorial Theory, 6:102–105, 1969.MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    T. Prellberg and R. Brak. Critical exponents from nonlinear functional equations for partially directed cluster models. J. Stat. Phys., 78(3/4):701–730, 1995.MATHCrossRefADSGoogle Scholar
  45. 45.
    V. Privman and M. Barma. Radii of gyration of fully and partially directed animals. Z. Phys. B: Cond. Mat., 57:59–63, 1984.CrossRefADSGoogle Scholar
  46. 46.
    V. Privman and N. M. Švrakić. Exact generating function for fully directed compact lattice animals. Phys. Rev. Lett., 60(12):1107–1109, 1988.CrossRefADSMathSciNetGoogle Scholar
  47. 47.
    H. Prodinger. The kernel method: a collection of examples. S ém. Lothar. Combin., 50:Art. B50f, 19 pp. (electronic), 2003/04.Google Scholar
  48. 48.
    R. C. Read. Contributions to the cell growth problem. Canad. J. Math., 14:1–20, 1962.MATHMathSciNetCrossRefGoogle Scholar
  49. 49.
    A. Rechnitzer. Haruspicy 2: the anisotropic generating function of self-avoiding polygons is not D-finite. J. Combin. Theory Ser. A, 113(3):520–546, 2006.MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    A. Salomaa and M. Soittola. Automata-theoretic aspects of formal power series. Springer-Verlag, New York, 1978. Texts and Monographs in Computer Science.Google Scholar
  51. 51.
    H. N. V. Temperley. Combinatorial problems suggested by the statistical mechanics of domains and of rubber-like molecules. Phys. Rev. (2), 103:1–16, 1956.MATHCrossRefMathSciNetADSGoogle Scholar
  52. 52.
    G. X. Viennot. Heaps of pieces. I. Basic definitions and combinatorial lemmas. In Combina-toire énumérative (Montré al, 1985), volume 1234 of Lecture Notes in Math., pages 321–350. Springer, Berlin, 1986.CrossRefGoogle Scholar
  53. 53.
    M. Vöge and A. J. Guttmann. On the number of hexagonal polyominoes. Theoret. Comput. Sci, 307(2):433–453, 2003.MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    T. Yuba and M. Hoshi. Binary search networks: a new method for key searching. Inform. Process. Lett., 24:59–65, 1987.MATHCrossRefGoogle Scholar
  55. 55.
    D. Zeilberger. Symbol-crunching with the transfer-matrix method in order to count skinny physical creatures. Integers, pages A9, 34pp. (electronic), 2000.Google Scholar
  56. 56.
    D. Zeilberger. The umbral transfer-matrix method. III. Counting animals. New York J. Math., 7:223–231 (electronic), 2001.MATHMathSciNetGoogle Scholar

Copyright information

© Canopus Academic Publishing Limited 2009

Authors and Affiliations

  • Mireille Bousquet-Mélou
    • 1
  • Richard Brak
    • 2
  1. 1.CNRS, LaBRI, Université Bordeaux 133405 Talence CedexFrance
  2. 2.Department of Mathematics and StatisticsThe University of MelbourneVictoriaAustralia

Personalised recommendations