Asymptotic analysis of expectations of plane partition statistics

  • Ljuben Mutafchiev


Assuming that a plane partition of the positive integer n is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as n becomes large. The generating functions that arise in this study are of the form Q(x)F(x), where \(Q(x)=\prod _{j=1}^\infty (1-x^j)^{-j}\) is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function F(x) around \(x=1\). The representation of a plane partition as a solid diagram of volume n allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part, the number of columns, the number of rows (that is, the three dimensions of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner et al. (Comb Probab Comput 23:1057–1086, 2014) related to linear integer partition statistics. We base our study on the Hayman’s method for admissible power series.


Plane partition statistic Asymptotic behavior Expectation 

Mathematics Subject Classification

05A17 05A16 11P82 



I am grateful the referee for the carefully reading the paper and for her/his helpful comments.


  1. 1.
    Abramovitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover Publ., Inc., New York (1965)Google Scholar
  2. 2.
    Andrews, G.E.: The Theory of Partitions. Encyclopedia Math. Appl. 2. Addison-Wesley, Reading, MA (1976)Google Scholar
  3. 3.
    Bodini, O., Fusy, E., Pivoteau, C.: Random sampling of plane partitions. Comb. Probab. Comput. 19, 201–226 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cerf, R., Kenyon, R.: The low of temperature expansion of the Wulff crystal in the \(3D\) Ising model. Commun. Math. Phys. 222, 147–179 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cohn, H., Larsen, M., Propp, J.: The shape of a typical boxed plane partition. New York J. Math. 4, 137–165 (1998)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Finch, S.: Mathematical Constants. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  8. 8.
    Freiman, G.A., Granovsky, B.L.: Asymptotic formula for a partition function of reversible coagulation-fragmentation processes. Isr. J. Math. 130, 259–279 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grabner, P., Knopfmacher, A., Wagner, S.: A general asymptotic scheme for the analysis of partition statistics. Comb. Probab. Comput. 23, 1057–1086 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Granovsky, B.L., Stark, D., Erlihson, M.: Meinardus’ theorem on weighted partitions: extemsions and a probailistic proof. Adv. Appl. Math. 41, 307–328 (2008)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hayman, W.K.: A generalization of Stirling’s formula. J. Reine Angew. Math. 196, 67–95 (1956)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Kamenov, E.P., Mutafchiev, L.R.: The limiting distribution of the trace of a random plane partition. Acta Math. Hung. 117, 293–314 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Krattenhaler, C.: Another involution principle—free bijective proof of Stanley’s hook-content formula. J. Comb. Theory Ser. A 88, 66–92 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    MacMahon, P.A.: Memoir on theory of partitions of numbers VI: partitions in two-dimensional space, to which is added an adumbration of the theory of partitions in three-dimensional space. Philos. Trans. R. Soc. London Ser. A 211, 345–373 (1912)CrossRefzbMATHGoogle Scholar
  15. 15.
    MacMahon, P.A.: Combinatory Analysis, Vol. 2. Cambridge University Press, Cambridege (1916); reprinted by Cheksea, New York (1960)Google Scholar
  16. 16.
    Meinardus, G.: Asymptotische aussagen über partitionen. Math. Z. 59, 388–398 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mutafchiev, L.: The size of the largest part of random plane partitions of large integers. Integers: Electr. J. Comb. Number Theory 6, A13 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Mutafchiev, L.: The size of the largest part of random weighted partitions of large integers. Comb. Probab. Comput. 22, 433–454 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mutafchiev, L.: An asymptotic scheme for analysis of expectations of plane partition statistics. Electron. Notes Discret. Math. 61, 893–899 (2017)CrossRefzbMATHGoogle Scholar
  20. 20.
    Mutafchiev, L., Kamenov, E.: Asymptotic formula for the number of plane partitions of positive integers. C. R. Acad. Bulgare Sci. 59, 361–366 (2006)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Okounkov, A., Reshetikhin, N.: Correlation function of Schur process with applications to local geometry of a random \(3\)-dimensional Young diagram. J. Am. Math. Soc. 16, 581–603 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pak, I.: Hook length formula and geometric combinatorics. Séminaire Lotharingen de Combinatoire 46, B46f (2001/02)Google Scholar
  23. 23.
    Pittel, B.: On dimensions of a random solid diagram. Comb. Probab. Comput. 14, 873–895 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Stanley, R.P.: Theory and applications of plane partitions I, II. Stud Appl. Math. 50(156–188), 259–279 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Stanley, R.P.: The conjugate trace and trace of a plane partition. J. Comb. Theory Ser. A 14, 53–65 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Stanley, R.P.: Enumerative Combinarics 2. Vol. 62 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1999)Google Scholar
  27. 27.
    Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1927)zbMATHGoogle Scholar
  28. 28.
    Wright, E.M.: Asymptotic partition formulae, I: plane partitions. Quart. J. Math. Oxford Ser. 2(2), 177–189 (1931)CrossRefzbMATHGoogle Scholar
  29. 29.
    Young, A.: On quantitative substitutional analysis. Proc. Lond. Math. Soc. 33, 97–146 (1901)MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.American University in BulgariaBlagoevgradBulgaria
  2. 2.Institute of Mathematics and Informatics of the Bulgarian Academy of SciencesSofiaBulgaria

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