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 $$\Delta _2(n)$$ .