Ramanujan-type congruences for broken 2-diamond partitions modulo 3

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. Andrews and Paule also posed three conjectures on the congruences of Delta(2)(n) modulo 2, 5 and 25. Hirschhorn and Sellers proved the conjectures for modulo 2, and Chan proved the two cases of modulo 5. For the case of modulo 3, Radu and Sellers obtained an infinite family of congruences for Delta(2)(n). In this paper, we obtain two infinite families of congruences for Delta(2)(n) modulo 3 based on a formula of Radu and Sellers, a 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 Delta(2)(243n + 142) a parts per thousand Delta(2)(243n + 223) a parts per thousand 0 (mod 3). The infinite family of Radu and Sellers and the two infinite families derived in this paper have two congruences in common, namely, Delta(2)(27n + 16) a parts per thousand Delta(2)(27n + 25) a parts per thousand 0 (mod 3).