Generalized congruence properties of the restricted partition function p(n,m)

Ramanujan-type congruences for the unrestricted partition function p(n) are well known and have been studied in great detail. The existence of Ramanujan-type congruences are virtually unknown for p(n,m), the closely related restricted partition function that enumerates the number of partitions of n into exactly m parts. Let a"" be any odd prime. In this paper we establish explicit Ramanujan-type congruences for p(n,a"") modulo any power of that prime a"" (alpha) . In addition, we establish general congruence relations for p(n,a"") modulo a"" (alpha) for any n.