Distinct partitions and overpartitions

In 1963, Peter Hagis, Jr. provided a Hardy-Ramanujan-Rademacher-type convergent series thatcan be used to compute an isolated value of the partition functionQ(n)which counts partitions ofninto distinctparts. ComputingQ(n)by this method requires arithmetic with very high-precision approximate real numbersand it is complicated. In this paper, we investigate new connections between partitions into distinct parts andoverpartitions and obtain a surprising recurrence relation for the number of partitions ofninto distinct parts.By particularization of this relation, we derive two different linear recurrence relations for the partition functionQ(n). One of them involves the thrice square numbers and the other involves the generalized octagonal num-bers. The recurrence relation involving the thrice square numbers provide a simple and fast computation of thevalue ofQ(n). This method uses only (large) integer arithmetic and it is simpler to program. Infinite families oflinear inequalities involving partitions into distinct parts and overpartitions are introduced in this context.