On $$(\ell , m)$$ ( ℓ , m ) -regular partitions with distinct parts

Abstract

Let $$a_{\ell ,m}(n)$$ denote the number of $$(\ell ,m)$$ -regular partitions of a positive integer n into distinct parts, where $$\ell $$ and m are relatively primes. In this paper, we establish several infinite families of congruences modulo 2 for $$a_{3,5}(n)$$ . For example, $$\begin{aligned} a_{3, 5}\left(2^{6\alpha +4}5^{2\beta }n+\frac{ 2^{6\alpha +3}5^{2\beta +1}-1}{3}\right) \equiv 0 , \end{aligned}$$ where $$\alpha , \beta \ge 0$$ .