The Ramanujan Journal

, Volume 1, Issue 1, pp 25–34

Divisibility of Certain Partition Functions by Powers of Primes

  • Basil Gordon
  • Ken Ono
Article

Abstract

Let \(k = p_1^{a_1 } p_2^{a_2 } \cdot \cdot \cdot p_m^{a_m } \) be the prime factorization of a positive integer k and let bk(n) denote the number of partitions of a non-negative integer n into parts none of which are multiples of k. If M is a positive integer, let Sk(N; M) be the number of positive integers ≤ N for which bk(n)≡ 0(modM). If \(p_i^{a_i } \geqslant \sqrt k \) we prove that, for every positive integer j\(\mathop {\lim }\limits_{N \to \infty } \frac{{S_k (N;p_i^j )}} {N} = 1. \) In other words for every positive integer j,bk(n) is a multiple of \(p_i^j \) for almost every non-negative integer n. In the special case when k=p is prime, then in representation-theoretic terms this means that the number ofp -modular irreducible representations of almost every symmetric groupSn is a multiple of pj. We also examine the behavior of bk(n) (mod \(p_i^j \)) where the non-negative integers n belong to an arithmetic progression. Although almost every non-negative integer n≡ (mod t) satisfies bk(n) ≡ 0 (mod \(p_i^j \)), we show that there are infinitely many non-negative integers nr (mod t) for which bk(n) ≢ 0 (mod \(p_i^j \)) provided that there is at least one such n. Moreover the smallest such n (if there are any) is less than 2 \(\cdot 10^8 p_i^{a_i + j - 1} k^2 t^4 \).

partitions congruences 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K. Alladi, “Partition identities involving gaps and weights,” Trans. Amer. Math. Soc. (to appear).Google Scholar
  2. 2.
    G. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, 1976, vol. 2.Google Scholar
  3. 3.
    A. Balog, H. Darmon, and K. Ono, “Congruences for Fourier coefficients of half integral weight modular forms and special values of L-functions,” Proceedings for the Conference in Honor of H. Halberstam, 1(1996), 105-128.Google Scholar
  4. 4.
    A. Biagioli, “The construction of modular forms as products of transforms of the Dedekind Eta function,” Acta. Arith. 54(1990), 273-300.Google Scholar
  5. 5.
    F. Garvan, “Some congruence properties for partitions that are p-cores,” Proc. London Math. Soc. 66(1993), 449-478.Google Scholar
  6. 6.
    A. Granville and K. Ono, “Defect zero p-blocks for finite simple groups,” Trans. Amer. Math. Soc. 348(1) (1996), 331-347.Google Scholar
  7. 7.
    G. James and A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, 1979.Google Scholar
  8. 8.
    N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer-Verlag, New York, 1984.Google Scholar
  9. 9.
    G. Ligozat, “Courbes modulaires de genre 1,” Bull. Soc. Math. France [Memoire 43] (1972), 1-80.Google Scholar
  10. 10.
    K. Ono, “On the positivity of the number of partitions that are t-cores,” Acta Arith. 66(3) (1994), 221-228.Google Scholar
  11. 11.
    K. Ono, “A note on the number of t-core partitions,” The Rocky Mtn. J. Math. 25(3) (1995), 1165-1169.Google Scholar
  12. 12.
    K. Ono, “Parity of the partition function,” Electronic Research Annoucements of the Amer. Math. Soc. 1(1) 35-42.Google Scholar
  13. 13.
    K. Ono, “Parity of the partition function in arithmetic progressions,” J. Reine Ange. Math., 472(1996), 1-15.Google Scholar
  14. 14.
    T.R. Parkin and D. Shanks, “On the distribution of parity in the partition function,” Math. Comp. 21(1967), 466-480.Google Scholar
  15. 15.
    J.-P. Serre, “Divisibilite des coefficients des formes modulaires de poids entier,” C.R. Acad. Sci. Paris A 279(1974), 679-682.Google Scholar
  16. 16.
    J. Sturm, “On the congruence of modular forms,” Springer Lect. Notes in Math. 1240, Springer Verlag, New York, 1984, pp. 275-280.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Basil Gordon
  • Ken Ono

There are no affiliations available

Personalised recommendations