The Ramanujan Journal

, Volume 1, Issue 1, pp 25–34

Divisibility of Certain Partition Functions by Powers of Primes


  • Basil Gordon
  • Ken Ono

DOI: 10.1023/A:1009711020492

Cite this article as:
Gordon, B. & Ono, K. The Ramanujan Journal (1997) 1: 25. doi:10.1023/A:1009711020492


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 \(\cdot 10^8 p_i^{a_i + j - 1} k^2 t^4 \).

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© Kluwer Academic Publishers 1997