Advertisement

The Ramanujan Journal

, Volume 30, Issue 3, pp 425–436 | Cite as

Generalized congruence properties of the restricted partition function p(n,m)

  • Brandt Kronholm
Article

Abstract

Ramanujan-type congruences for the unrestricted partition function p(n) are well known and have been studied in great detail. The existence of Ramanujan-type congruences are virtually unknown for p(n,m), the closely related restricted partition function that enumerates the number of partitions of n into exactly m parts. Let be any odd prime. In this paper we establish explicit Ramanujan-type congruences for p(n,) modulo any power of that prime α . In addition, we establish general congruence relations for p(n,) modulo α for any n.

Keywords

Partition Congruence Generating function Ramanujan 

Mathematics Subject Classification

05A17 11P83 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahlgren, S.: Distribution of the partition function modulo composite integers M. Math. Ann. 318, 795–803 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Andrews, G.E.: The Theory of Partitions, the Encyclopedia of Mathematics and Its Applications Series. Addison-Wesley, New York (1976). Reissued, Cambridge University Press, New York (1998) Google Scholar
  3. 3.
    Atkin, A.O.L.: Congruence Heke Operators. Proc. Symp. Pure Math. 12, 33–40 (1969) CrossRefGoogle Scholar
  4. 4.
    Atkin, A.O.L.: Proof of a conjecture of Ramanujan. Glascow Math. J. 8, 14–32 (1967) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Gupta, H.: Partitions—a survey. J. Res. Natl. Bureau Stand. Math. Sci. 74B(1) (1970) Google Scholar
  6. 6.
    Gupta, H., Gwyther, E.E., Miller, J.C.P.: Tables of Partitions. Royal Soc. Math. Tables, vol. 4. Cambridge University Press, Cambridge (1958) zbMATHGoogle Scholar
  7. 7.
    Kronholm, B.: On Congruence Properties of p(n,m). PAMS 133, 2891–2895 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Kronholm, B.: On consecutive congruence properties of p(n,m). Integers 7, #A16 (2007) MathSciNetGoogle Scholar
  9. 9.
    Kwong, Y.H.: Minimum periods of binomial coefficients modulo M. Fibonacci Q., 27, 348–351 (1989) MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kwong, Y.H.: Minimum periods of partition functions modulo M. Util. Math. 35, 3–8 (1989) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kwong, Y.H.: Periodicities of a class of infinite integer sequences modulo M. J. Number Theory 31, 64–79 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Nijenhuis, A., Wilf, H.S.: Periodicities of partition functions and Stirling numbers modulo p. J. Number Theory 25, 308–312 (1987) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Ono, K.: Distribution of the partition function modulo m. Ann. Math. 151, 293–307 (2000) zbMATHCrossRefGoogle Scholar
  14. 14.
    Ramanujan, S.: Congruence properties of partitions. Proc. Lond. Math. Soc. (2) 19, 207–210 (1919) zbMATHGoogle Scholar
  15. 15.
    Ramanujan, S.: Collected Papers. Cambridge University Press, London (1927). Reprinted: AMS, Chelsea (2000) with new preface and extensive commentary by B. Berndt zbMATHGoogle Scholar
  16. 16.
    Sylvester, J.J.: On subinvariants, i.e. semi-invariants to binary quantics of an unlimited order. With an excursus on rational fractions and partitions. Am. J. Math. 5, 79–136 (1882) MathSciNetCrossRefGoogle Scholar
  17. 17.
    Watson, G.N.: Ramanujans Vermutungüber Zerfällungsanzahlen. J. Reine Angew. Math. 179, 97–128 (1938) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Whittier CollegeWhittierUSA

Personalised recommendations