Congruences Modulo 2 for Certain Partition Functions

Abstract

Let $b_{3,5}(n)$ denote the number of partitions of $n$ into parts that are not multiples of 3 or 5. We establish several infinite families of congruences modulo 2 for $b_{3,5}(n)$. In the process, we also prove numerous parity results for broken 7-diamond partitions.

Infinite families of congruences modulo $2$ for $(\ell, k)$-regular partitions

“…In [MHS16], Naika, Hemanthkumar and Bharadwaj have proved infinite families of congruences modulo 2 for b 3,5 (n)-regular partition. For example, they showed that [MHS16, Theorem 1.1] if p is an odd prime −15 p L = 1, 1 ≤ i ≤ p − 1 and j, n ≥ 0, b 3,5 2 × p 2j+2 n + (6i + 2p) × p 2j+1 + 1 3 ≡ 0 (mod 2).…”