On the Distribution of Multiplicities in Integer Partitions

Abstract

We study the distribution of the number of parts of given multiplicity (or equivalently, ascents of given size) in integer partitions. In this paper we give methods to compute asymptotic formulas for the expected value and variance of the number of parts of multiplicity d (d is a positive integer) in a random partition of a large integer n and also prove that the limiting distribution is asymptotically normal for fixed d. However, if we let d increase with n, we get a phase transition for d around n 1/4. Our methods can also be applied to the so−called λ-partitions where the parts are members of a sequence of integers λ.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Andrews G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  2. 2.

    Brennan C., Knopfmacher A., Wagner S.: The distribution of ascents of size d or more in partitions of n. Combin. Probab. Comput. 17(4), 495–509 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Corteel S., Pittel B., Savage C.D., Wilf H.S.: On the multiplicity of parts in a random partition. Random Structures Algorithms 14(2), 185–197 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Curtiss J.H.: A note on the theory of moment generating functions. Ann. Math. Statistics 13, 430–433 (1942)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Erdös P., Lehner J.: The distribution of the number of summands in the partitions of a positive integer. Duke Math. J. 8, 335–345 (1941)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Flajolet P., Gourdon X., Dumas P.: Mellin transforms and asymptotics: harmonic sums. Theoret. Comput. Sci. 144(1-2), 3–58 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Flajolet P., Sedgewick R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)

    MATH  Book  Google Scholar 

  8. 8.

    Goh W.M.Y., Schmutz E.: The number of distinct part sizes in a random integer partition. J. Combin. Theory Ser. A 69(1), 149–158 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Grabner, P.J., Knopfmacher, A., Wagner, S.: A general asymptotic scheme for moments of partition statistics. Preprint

  10. 10.

    Hwang H.-K.: Limit theorems for the number of summands in integer partitions. J. Combin. Theory Ser. A 96(1), 89–126 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Knopfmacher A., Munagi A.O.: Successions in integer partitions. Ramanujan J. 18(3), 239–255 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Madritsch M., Wagner S.: A central limit theorem for integer partitions. Monatsh. Math. 161(1), 85–114 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  13. 13.

    Ralaivaosaona D.: On the number of summands in a random prime partition. Monatsh. Math. 166(3), 505–524 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Roth, K.F., Szekeres, G.: Some asymptotic formulae in the theory of partitions. Quart. J. Math. Oxford Ser. (2) 5, 241–259 (1954)

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Dimbinaina Ralaivaosaona.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ralaivaosaona, D. On the Distribution of Multiplicities in Integer Partitions. Ann. Comb. 16, 871–889 (2012). https://doi.org/10.1007/s00026-012-0165-2

Download citation

Mathematics Subject Classification

  • 05A17
  • 11P82

Keywords

  • integer partitions
  • multiplicities
  • ascents
  • asymptotic expansions
  • limit distribution