Ramanujan-type Congruences for Broken 2-Diamond Partitions Modulo 3

Abstract

The notion of broken $k$-diamond partitions was introduced by Andrews and Paule. Let $Δ_k(n)$ denote the number of broken k-diamond partitions of $n$. They also posed three conjectures on the congruences of $Δ_2(n)$ modulo 2, 5 and 25. Hirschhorn and sellers proved the conjectures for modulo 2, and Chan proved cases of modulo 5. For the case of modulo 3, Radu and Sellers obtained an infinite family of congruences for $Δ_2(n)$. In this paper, we obtain two infinite families of congruences for $Δ_2(n)$ modulo 3 based on a formula of Radu and Sellers, the 3-dissection formula of the generating function of triangular number due to Berndt, and the properties of the $U$-operator, the $V$-operator, the Hecke operator and the Hecke eigenform. For example, we find that $Δ_2(243n+142)\equiv Δ_2(243n+223)\equiv0\pmod{3}$. The infinite family of Radu and Sellers and the two infinite families derived in this paper have two congruences in common, namely, $Δ_2(27n+16)\equivΔ_2(27n+25)\equiv0 \pmod{3}$.