Congruence properties modulo powers of 2 for partition pairs into distinct parts

Abstract

Let $Q(n)$ denote the number of partitions of $n$ into distinct parts. In 1997, Gordon and Ono proved that almost all values of $Q(n)$ are divisible by $2^m$ with any fixed positive integer $m$. Let $Q_2(n)$ denote the number of partition pairs of $n$ into distinct parts. A result derived by Ray and Barman reveals that almost all values of $Q_2(n)$ are also divisible by $2^m$ with any fixed positive integer $m$. Quite recently, the author derived several internal congruences and congruences modulo powers of $2$ satisfied by $Q(n)$. In this paper, we prove some internal congruences and congruences modulo powers of 2 for $Q_2(n)$. Moreover, we prove an infinite family of congruence relations modulo $4$ and dozens of congruence relations modulo powers of $2$ enjoyed by $Q_2(n)$. Finally, we pose two conjectures on congruence properties modulo powers of $2$ for $Q_2(n)$.