Divisibility of Certain Partition Functions by Powers of Primes

Let be the prime factorization of a positive integer k and let b_k(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 S_k(N; M) be the number of positive integers N for which b_k(n) 0(modM). If we prove that, for every positive integer j In other words for every positive integer j,b_k(n) is a multiple of 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 groupS_n is a multiple of p^j. We also examine the behavior of b_k(n) (mod ) where the non-negative integers n belong to an arithmetic progression. Although almost every non-negative integer n (mod t) satisfies b_k(n) 0 (mod ), we show that there are infinitely many non-negative integers n r (mod t) for which b_k(n) 0 (mod ) provided that there is at least one such n. Moreover the smallest such n (if there are any) is less than 2 .