Congruences modulo 4 for broken k-diamond partitions

Abstract

The notion of broken k-diamond partitions was introduced by Andrews and Paule. Let $\Delta _k(n)$ denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, a number of parity results satisfied by $\Delta _k(n)$ for small values of k have been proved by Radu and Sellers and others. However, congruences modulo 4 for $\Delta _k(n)$ are unknown. In this paper, we will prove five congruences modulo 4 for $\Delta _5(n)$, four infinite families of congruences modulo 4 for $\Delta _7(n)$ and one congruence modulo 4 for $\Delta _{11}(n)$ by employing theta function identities. Furthermore, we will prove a new parity result for $\Delta _2(n)$.