# Estimating the Asymptotics of Solid Partitions

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## Abstract

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.

## Keywords

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

### Acknowledgments

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.

## References

- 1.Andrews, G.E.: The Theory of Partitions, vol. 2. Cambridge University Press, Cambridge (1998)MATHGoogle Scholar
- 2.Almkvist, G.: A rather exact formula for the number of plane partitions. Contemp. Math.
**143**, 21–26 (1993)CrossRefMathSciNetGoogle Scholar - 3.Almkvist, G.: Asymptotic formulas and generalized Dedekind sums. J. Exp. Math.
**7**, 343–359 (1998)CrossRefMATHMathSciNetGoogle Scholar - 4.Govindarajan, S., Prabhakar, N.S.: A superasymptotic formula for the number of plane partitions, arXiv preprint arXiv:1311.7227 (2013)
- 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.Wright, E.M.: Asymptotic partition formulae I. Plane partitions. Q. J. Math. Oxford, Ser.
**2**, 177–189 (1931)CrossRefGoogle Scholar - 7.Govindarajan, S., Balakrishnan, S.: The solid partitions project, http://boltzmann.wikidot.com/solid-partitions
- 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.MacMahon, P.A.: Combinatory Analysis. Cambridge University Press, Cambridge (1916)Google Scholar
- 10.Levine, D., Steinhardt, P.J.: Quasicrystals: a new class of ordered structures. Phys. Rev. Lett.
**53**, 2477–2480 (1984)CrossRefADSGoogle Scholar - 11.Elser, V.: Comment on “Quasicrystals: A New Class of Ordered Structures”. Phys. Rev. Lett.
**54**, 1730 (1985)CrossRefADSGoogle Scholar - 12.Mosseri, R., Bailly, F.: Configurational entropy in octagonal tiling models. Int. J. Mod. Phys. B
**7**, 1427–1436 (1993)CrossRefADSMATHMathSciNetGoogle Scholar - 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.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.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.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.Hutchinson, M., Widom, M.: Enumeration of octagonal tilings, arXiv:1306.5977 [math.CO] (2013)
- 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.Gopakumar, R., Vafa, C.: M-Theory and Topological Strings-I, ariXiv:hep-th/9809187 (1998)
- 20.Gopakumar, R., Vafa, C.: M-Theory and Topological Strings-II, arXiv:hep-th/9812127 (1998)
- 21.Behrend, K., Bryan, J., Szendröi, B.: Motivic degree zero Donaldson–Thomas invariants. Invent. Math.
**192**, 111–160 (2013)CrossRefADSMATHMathSciNetGoogle Scholar - 22.Balakrishnan, S., Govindarajan, S., Prabhakar, N.S.: On the asymptotics of higher-dimensional partitions. J. Phys. A
**45**, 055001 (2012)CrossRefADSMathSciNetGoogle Scholar - 23.Bratteli, O.: Inductive limits of finite dimensional C*-algebras. Trans. Am. Math. Soc.
**171**, 195–234 (1972)MATHMathSciNetGoogle Scholar - 24.Sagan, B.E.: The Symmetric Group. Wadsworth and Brooks/Cole, Pacific Grove (1991)MATHGoogle Scholar
- 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.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.Bratley, P., McKay, J.K.S.: Algorithm 313: Multi-dimensional partition generator. Commun. ACM
**10**, 666 (1967)CrossRefGoogle Scholar - 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.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.Kenyon, R., Okounkov, A., Sheffield, S.: Dimers and Amoebae. Ann. Math.
**163**, 1019–1056 (2006)CrossRefMATHMathSciNetGoogle Scholar - 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.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.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.Destainville, N.: Entropy and boundary conditions in random rhombus tilings. J. Phys. A
**31**, 6123–6139 (1998)CrossRefADSMATHMathSciNetGoogle Scholar - 35.Björner, A., Stanley, R.P.: A combinatorial miscellany, L’enseignement mathématique, Monograph no. 42, Genève, (2010)Google Scholar
- 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