Distinct partitions and overpartitions

Abstract

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

Efficient computation of the overpartition function and applications

“…Overpartitions have also been studied in relation to topics of great interest in number theory and the theory of integer partitions, like ranks, cranks and mock theta functions, for example in [BLO09], [ADSY17], [Lin20] and [Zha21], just to cite a few. More recently, overpartitions have been linked directly to the number of partitions of an integer into different parts in [Mer22].…”