Abstract
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 \(\mathfrak{B}_{7^\alpha}(n)\).
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Yu, JJ. New Congruences for Broken \(k\)-Diamond and \(k\) Dots Bracelet Partitions. Math Notes 112, 393–405 (2022). https://doi.org/10.1134/S0001434622090085
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DOI: https://doi.org/10.1134/S0001434622090085