The enumeration of self-avoiding polygons, and other families of lattice animals, is one of the most famous problems in enumerative combinatorics, and despite many years of intensive study these problems remain completely open.
Let p n be the number of self-avoiding polygons on the square lattice of perimeter 2n and let \(G(z) = \sum {p_n z^n } \) be the corresponding generating function. Neither an explicit nor a useful implicit expression is known for either the G(z) or p n . The most efficient means of computing p n is the finite-lattice method (see Chapter 7). This method is exponentially faster than brute-force enumeration (and considerably so!), but still requires exponential time and space.
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A Maple package developed by Bruno Salvy, Paul Zimmermann and Eithne Murray at INRIA, France. Available from http://algo.inria.fr/libraries/ at time of printing.
D. Bennett-Wood, J.L. Cardy, I.G. Enting, A.J. Guttmann, and A.L. Owczarek. On the non-universality of a critical exponent for self-avoiding walks. Nuclear Physics B, 528(3):533–552, 1998.
M. Bousquet-Mélou. A method for the enumeration of various classes of column-convex polygons. Discrete Math, 154(1–3):1–25, 1996.
M. Bousquet-Mélou, A.J. Guttmann, W.P. Orrick, and A. Rechnitzer. Inversion relations, reciprocity and polyominoes. Annals of Combinatorics, 3(2):223–249, 1999.
M. Bousquet-Mélou and A. Rechnitzer. Lattice animals and heaps ofdimers. Discrete Mathematics, 258:235–274, 2002.
M. Bousquet-Mélou and A. Rechnitzer. The site-perimeter of bargraphs. Advances in Applied Mathematics, 31(1):86–112, 2003.
A.J. Guttmann. Asymptotic analysis of power-series expansions. In C. Domb and J.L. Lebowitz, editors, Phase Transitions and Critical Phenomena, volume 13, pages 1–234. Academic, New York, 1989.
A.J. Guttmann and I.G. Enting. Solvability of Some Statistical Mechanical Systems. Physical Review Letters, 76(3):344–347, 1996.
I. Jensen. Anisotropic series for site-animals. Personal communication with the author.
I. Jensen and A.J. Guttmann. Self-avoiding polygons on the square lattice. J. Phys. A: Math. Gen, 32:4867–76, 1999.
M. Klazar. Non-P-recursiveness of numbers of matchings or linear chord diagrams with many crossings. Advances in Applied Mathematics, 30(1–2):126–136, 2003.
L. Lipshitz. D-finite power series. J. Algebra, 122(2):353–373, 1989.
J. Noonan and D. Zeilberger. The Enumeration of Permutations with a Prescribed Number of Forbidden Patterns. Advances in Applied Mathematics, 17(4):381–407, 1996.
A. Rechnitzer. The anisotropic generating functions of bond-animals and bond trees are not D-finite. Unpublished.
A. Rechnitzer. Haruspicy and anisotropic generating functions. Advances in Applied Mathematics, 30(1–2):228–257, 2003.
A. Rechnitzer. Haruspicy 2: The anisotropic generating function of self-avoiding polygons is not D-finite. Journal of Combinatorial Theory Series A, 113(3):520–546, 2006.
A. Rechnitzer. Haruspicy 3: the anisotropic generating function of directed bond-animals is not D-finite. Journal of Combinatorial Theory Series A, 113(6):1031–1049, 2006.
A. N. Rogers. On the anisotropic generating function of self-avoiding walks. Personal com-muncation with author—work formed part of ANR's PhD thesis submitted to The University of Melbourne, 2004.
B. Salvy and P. Zimmerman. GFUN: a Maple package for the manipulation of generating and holonomic functions in one variable. ACM Transactions on Mathematical Software, 20(2):163–177, 1994.
R.P. Stanley. Enumerative Combinatorics, volume 1. Cambridge University Press, Cambridge, 1996.
H.N. Temperley. Combinatorial Problems Suggested by the Statistical Mechanics of Domains and of Rubber-Like Molecules. Physical Review, 103(1):1–16, 1956.
E.T. Whittaker and G.N. Watson. A Course of Modern Analysis. Cambridge University Press, Cambridge, 1996.
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Rechnitzer, A. (2009). The Anisotropic Generating Function of Self-Avoiding Polygons is not D-Finite. In: Guttman, A.J. (eds) Polygons, Polyominoes and Polycubes. Lecture Notes in Physics, vol 775. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9927-4_5
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