# Divisibility of Certain Partition Functions by Powers of Primes

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## 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 of*p* -modular irreducible representations of almost every symmetric group*S*_{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 \(\cdot 10^8 p_i^{a_i + j - 1} k^2 t^4 \).

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