Journal of Statistical Physics

, Volume 158, Issue 4, pp 950–967 | Cite as

Estimating the Asymptotics of Solid Partitions

  • Nicolas Destainville
  • Suresh Govindarajan


We study the asymptotic behavior of solid partitions using transition matrix Monte Carlo simulations. If \(p_3(n)\) denotes the number of solid partitions of an integer \(n\), we show that \(\lim _{n\rightarrow \infty } n^{-3/4} \log p_3(n)\sim 1.822\pm 0.001\). This shows clear deviation from the value \(1.7898\), attained by MacMahon numbers \(m_3(n)\), that was conjectured to hold for solid partitions as well. In addition, we find estimates for other sub-leading terms in \(\log p_3(n)\). In a pattern deviating from the asymptotics of line and plane partitions, we need to add an oscillatory term in addition to the obvious sub-leading terms. The period of the oscillatory term is proportional to \(n^{1/4}\), the natural scale in the problem. This new oscillatory term might shed some insight into why partitions in dimensions greater than two do not admit a simple generating function.


Solid partitions of an integer Asymptotic expansion Transition matrix Monte Carlo simulations 



SG would like to thank Intel India for financially supporting the numerical study of solid partitions. We also thank the High Performance Computing Environment at IIT Madras which provided us access to the Virgo and Vega super clusters where our Monte Carlo simulations were carried out.

Supplementary material


  1. 1.
    Andrews, G.E.: The Theory of Partitions, vol. 2. Cambridge University Press, Cambridge (1998)MATHGoogle Scholar
  2. 2.
    Almkvist, G.: A rather exact formula for the number of plane partitions. Contemp. Math. 143, 21–26 (1993)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Almkvist, G.: Asymptotic formulas and generalized Dedekind sums. J. Exp. Math. 7, 343–359 (1998)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Govindarajan, S., Prabhakar, N.S.: A superasymptotic formula for the number of plane partitions, arXiv preprint arXiv:1311.7227 (2013)
  5. 5.
    Mutafchiev, L., Kamenov, E.: Asymptotic formula for the number of plane partitions of positive integers. Compt. Rend. Acad. Bulg. Sci. 59, 361–366 (2006)MATHMathSciNetGoogle Scholar
  6. 6.
    Wright, E.M.: Asymptotic partition formulae I. Plane partitions. Q. J. Math. Oxford, Ser. 2, 177–189 (1931)CrossRefGoogle Scholar
  7. 7.
    Govindarajan, S., Balakrishnan, S.: The solid partitions project,
  8. 8.
    Mustonen, V., Rajesh, R.: Numerical estimation of the asymptotic behaviour of solid partitions of an integer. J. Phys. A 36, 6651–6659 (2003)CrossRefADSMathSciNetGoogle Scholar
  9. 9.
    MacMahon, P.A.: Combinatory Analysis. Cambridge University Press, Cambridge (1916)Google Scholar
  10. 10.
    Levine, D., Steinhardt, P.J.: Quasicrystals: a new class of ordered structures. Phys. Rev. Lett. 53, 2477–2480 (1984)CrossRefADSGoogle Scholar
  11. 11.
    Elser, V.: Comment on “Quasicrystals: A New Class of Ordered Structures”. Phys. Rev. Lett. 54, 1730 (1985)CrossRefADSGoogle Scholar
  12. 12.
    Mosseri, R., Bailly, F.: Configurational entropy in octagonal tiling models. Int. J. Mod. Phys. B 7, 1427–1436 (1993)CrossRefADSMATHMathSciNetGoogle Scholar
  13. 13.
    Destainville, N., Mosseri, R., Bailly, F.: Fixed-boundary octagonal random tilings: a combinatorial approach. J. Stat. Phys. 102, 147–190 (2001)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Widom, M., Mosseri, R., Destainville, N., Bailly, F.: Arctic octahedron in three-dimensional rhombus tilings and related integer solid partitions. J. Stat. Phys. 109, 945–965 (2002)CrossRefADSMATHMathSciNetGoogle Scholar
  15. 15.
    Destainville, N., Mosseri, R., Bailly, F.: A formula for the number of tilings of an octagon by rhombi. Theor. Comput. Sci. 319, 71–81 (2004)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Destainville, N., Widom, M., Mosseri, R., Bailly, F.: Random tilings of high symmetry: I. Mean-field theory. J. Stat. Phys. 120, 799–835 (2005)CrossRefADSMATHMathSciNetGoogle Scholar
  17. 17.
    Hutchinson, M., Widom, M.: Enumeration of octagonal tilings, arXiv:1306.5977 [math.CO] (2013)
  18. 18.
    Vidal, J., Destainville, N., Mosseri, R.: Quantum dynamics in high codimension tilings: from quasiperiodicity to disorder. Phys. Rev. B 68, 172202 (2003)CrossRefADSGoogle Scholar
  19. 19.
    Gopakumar, R., Vafa, C.: M-Theory and Topological Strings-I, ariXiv:hep-th/9809187 (1998)
  20. 20.
    Gopakumar, R., Vafa, C.: M-Theory and Topological Strings-II, arXiv:hep-th/9812127 (1998)
  21. 21.
    Behrend, K., Bryan, J., Szendröi, B.: Motivic degree zero Donaldson–Thomas invariants. Invent. Math. 192, 111–160 (2013)CrossRefADSMATHMathSciNetGoogle Scholar
  22. 22.
    Balakrishnan, S., Govindarajan, S., Prabhakar, N.S.: On the asymptotics of higher-dimensional partitions. J. Phys. A 45, 055001 (2012)CrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Bratteli, O.: Inductive limits of finite dimensional C*-algebras. Trans. Am. Math. Soc. 171, 195–234 (1972)MATHMathSciNetGoogle Scholar
  24. 24.
    Sagan, B.E.: The Symmetric Group. Wadsworth and Brooks/Cole, Pacific Grove (1991)MATHGoogle Scholar
  25. 25.
    Atkin, A.O.L., Bratley, P., MacDonald, I.G., McKay, K.S.: Some computations for m-dimensional partitions. Proc. Camb. Philos. Soc. 63, 1097–1100 (1967)CrossRefADSMATHMathSciNetGoogle Scholar
  26. 26.
    Bhatia, D.P., Prasad, M.A., Arora, D.: Asymptotic results for the number of multidimensional partitions of an integer and directed compact lattice animals. J. Phys. A 30, 2281–2285 (1997)CrossRefADSMATHMathSciNetGoogle Scholar
  27. 27.
    Bratley, P., McKay, J.K.S.: Algorithm 313: Multi-dimensional partition generator. Commun. ACM 10, 666 (1967)CrossRefGoogle Scholar
  28. 28.
    Erdos, P., Lehner, J.: The distribution of the number of summands in the partitions of a positive integer. Duke Math. 8, 335–345 (1941)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Cerf, R., Kenyon, R.: The low temperature expansion of the Wulff crystal in the 3D Ising model. Comm. Math. Phys. 222, 147–179 (2001)CrossRefADSMATHMathSciNetGoogle Scholar
  30. 30.
    Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and Amoebae. Ann. Math. 163, 1019–1056 (2006)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    de Bruijn, N.G.: Algebraic theory of Penrose’s nonperiodic tilings of the plane. I. Neder. Akad. Wetensch. Indag. Math. 43, 39–52 (1981)CrossRefMATHGoogle Scholar
  32. 32.
    de Bruijn, N.G.: Algebraic theory of Penrose’s nonperiodic tilings of the plane. II. Neder. Akad. Wetensch. Indag. Math. 43, 53–66 (1981)CrossRefMATHGoogle Scholar
  33. 33.
    Henley, C.L.: Random tiling models. In: Di Vincenzo, D.P., Steingart, P.J. (eds.) Quasicrystals, the State of the Art, p. 429. World Scientific, Singapore (1991)CrossRefGoogle Scholar
  34. 34.
    Destainville, N.: Entropy and boundary conditions in random rhombus tilings. J. Phys. A 31, 6123–6139 (1998)CrossRefADSMATHMathSciNetGoogle Scholar
  35. 35.
    Björner, A., Stanley, R.P.: A combinatorial miscellany, L’enseignement mathématique, Monograph no. 42, Genève, (2010)Google Scholar
  36. 36.
    Linde, J., Moore, C., Nordahl, M.G.: An \(n\)-dimensional generalization of the rhombus tiling. In: Proceedings of the 1st International conference on Discrete Models: Combinatorics, Computation, and Geometry (DM-CCG+01), M. Morvan, R. Cori, J. Mazoyer and R. Mosseri, eds., Discrete Math. Theo. Comp. Sc. AA:23 (2001)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.LPT (IRSAMC) Laboratoire de Physique Théorique, UPS CNRSUniversité de ToulouseToulouseFrance
  2. 2.Department of PhysicsIndian Institute of Technology MadrasChennaiIndia

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