New Congruences for Broken $k$-Diamond and $k$ Dots Bracelet Partitions

Let $\Delta_k(n)$ denote the number of broken $k$-diamond partitions of $n$. Recently, Radu and Sellers studied the parity of the function $\Delta_3(n)$ and posed a conjecture. They proved that the conjecture is true for $\alpha =1$. Using the theory of modular forms, we give a new proof of the conjecture for $\alpha = 1$. Based on these results, we deduce some new infinite families of congruences modulo 2 for $\Delta_3(n)$. Similarly, we find several new congruences modulo 4 for $\Delta_3(n)$ and a new Ramanujan type congruence for $\Delta_2(n)$ modulo 2. Furthermore, let $\mathfrak{B}_k(n)$ denote the number of $k$ dots bracelet partitions of $n$. We also deduce some new Ramanujan type congruences for $\mathfrak{B}_{5^\alpha}(n)$ and $\mathfrak{B}_{7^\alpha}(n)$.