Divisibility of Certain Partition Functions by Powers of Primes

Let k = p11p a2 2 · · ·p am 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 ≤ N for which bk(n) ≡ 0 (mod M). If pi i ≥ √ k, we prove that, for every positive integer j lim N→∞ Sk(N ; p j i ) N = 1. In other words for every positive integer j, bk(n) is a multiple of p j i 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 of p-modular irreducible representations of almost every symmetric group Sn is a multiple of pj . We also examine the behavior of bk(n) (mod p j i ) where the non-negative integers n belong to an arithmetic progression. Although almost every non-negative integer n ≡ r (mod t) satisfies bk(n) ≡ 0 (mod pji ), we show that there are infinitely many non-negative integers n ≡ r (mod t) for which bk(n) 6≡ 0 (mod pji ) provided that there is at least one such n. Moreover the smallest such n (if there are any) is less than 2 · 108pai+j−1 i k 2t4. 1. The main theorem A partition of a positive integer n is any non-increasing sequence of positive integers whose sum is n. The number of such partitions is denoted by p(n), and the number of partitions where the summands are distinct is denoted by q(n). If k is a positive integer, let bk(n) be the number of partitions of n into parts none which are multiples of k. It is known that b2(n) = q(n), and if p is prime, then bp(n) is the number of irreducible p−modular representations of the symmetric group Sn [7]. The generating function Gk(x) for bk(n) is given by the infinite product: