Divisibility of Certain Partition Functions by Powers of Primes

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 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(mod M). 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, b _k(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 groupS _n is a multiple of p ^j. We also examine the behavior of b _k(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 b _k(n) ≡ 0 (mod $p_i^j$), we show that there are infinitely many non-negative integers n≡ r (mod t) for which b _k(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 $\cdot 10^8 p_i^{a_i + j - 1} k^2 t^4$.