Abstract
We derive self-reciprocity properties for a number of polyomino generating functions, including several families of column-convex polygons, three-choice polygons, and staircase polygons with a staircase hole. In so doing, we establish a connection between the reciprocity results known to combinatorialists and the inversion relations used by physicists to solve models in statistical mechanics. For several classes of convex polygons, the inversion (reciprocity) relation, augmented by certain symmetry and analyticity properties, completely determines the anisotropic perimeter generating function.
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References
R.J. Baxter, Hard hexagons: Exact solution, J. Phys. A: Math. Gen.13 (1980) L61-L70.
R.J. Baxter, Exactly solved models, In: Fundamental Problems in Statistical Mechanics V, Proceedings of the 1980 Enschede Summer School, E.G.D. Cohen, Ed., North Holland, Amsterdam, 1980, pp. 109–141.
R.J. Baxter, Two-dimensional models in statistical mechanics, In: Statistical Mechanics and Field Theory, Proceedings of the Seventh Physics Summer School, ANU, 1994, V.V. Bazhanov and C.J. Burden, Eds., World Scientific, Singapore, 1995, pp. 129–167.
R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982.
R.J. Baxter, The inversion relation method for some two-dimensional exactly solved models in lattice statistics, J. Stat. Phys.28 (1982) 1–41.
M. Bousquet-Mélou, Convex polyominoes and heaps of segments, J. Phys. A: Math. Gen.25 (1992) 1925–1934.
M. Bousquet-Mélou, A method for the enumeration of various classes of column-convex polygons, Discrete Math.154 (1996) 1–25.
R. Brak, J.W. Essam, and A.L. Owczarek, From the Bethe Ansatz to the Gessel-Viennot Theorem, Ann. Combin.3 (1999) 251–263.
A. Conway, A.J. Guttmann and M. Delest, On the number of three-choice polygons, Math. Comput. Model26 (1997) 51–58.
J.M. Fédou, Fonctions de Bessel, empilements et tresses, In: Proceedings of the 4th Conference on Formal Power Series and Algebraic Combinatorics, P. Leroux and C. Reutenauer, Eds., Publications du LACIM, Université du Québec à Montréal, Vol. 11, 1992, pp. 189–202.
M.T. Jaekel and J.-M. Maillard, Symmetry relations in exactly soluble models, J. Phys. A: Math. Gen.15 (1982) 1309–1325.
M.T. Jaekel and J.-M. Maillard, Inverse functional relation on the Potts model, J. Phys. A: Math. Gen.15 (1982) 2241–2257.
A.J. Guttmann, I. Jensen, and I.G. Enting, Punctured polygons and polyominoes on the square lattice, J. Phys. A, submitted.
K.Y. Lin and S.J. Chang, Rigorous results for the number of convex polygons on the square and honeycomb lattices, J. Phys. A: Math. Gen.21 (1988) 2635–2642.
J.-M. Maillard, The inversion relation, J. Physique46 (1985) 329–341.
L. Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev.65 (1944) 117–149.
S.V. Pokrovsky and Yu.A. Bashilov, Star-triangle relations in the exactly solvable statistical models, Comm. Math. Phys.84 (1982) 103–132.
R. Shankar, Simple derivation of the Baxter-model free energy, Phys. Rev. Lett.47 (1981) 1177–1180.
C.L. Schultz, Solvableq-state models in lattice statistics and quantum field theory, Phys. Rev. Lett.46 (1981) 629–632.
R.P. Stanley, Combinatorial reciprocity theorems, Adv. Math.14 (1974) 194–253.
R.P. Stanley, Linear diophantine equations and local cohomology, Inv. Math.68 (1982) 175–193.
R.P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and Brooks/Cole, Monterey, 1986.
Yu.G. Stroganov, A new calculation method for partition functions in some lattice models, Phys. Lett.A74 (1979) 116–118.
H.N.V. Temperley, Combinatorial problems suggested by the statistical mechanics of domains and of rubber-like molecules, Phys. Rev.103 (1956) 1–16.
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Bousquet-Mélou, M., Guttmann, A.J., Orrick, W.P. et al. Inversion relations, reciprocity and polyominoes. Annals of Combinatorics 3, 223–249 (1999). https://doi.org/10.1007/BF01608785
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DOI: https://doi.org/10.1007/BF01608785