Parity Results for \(p\)-Regular Partitions with Distinct Parts

Abstract

We consider the partition function \(b’_p(n)\), which counts the number of partitions of the integer \(n\) into distinct parts with no part divisible by the prime \(p\). We prove the following: Let \(p\) be a prime greater than \(3\) and let \(r\) be an integer between \(1\) and \(p-1\), inclusively, such that \(24r+1\) is a quadratic nonresidue modulo \(p\). Then, for all nonnegative integers \(n\), \(b’_p{(pn+r)} \equiv 0 \pmod{2}.\)