, Volume 30, Issue 3, pp 425-436

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

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Abstract

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 be any odd prime. In this paper we establish explicit Ramanujan-type congruences for p(n,) modulo any power of that prime α . In addition, we establish general congruence relations for p(n,) modulo α for any n.