Formulas for partition k-tuples with t-cores

Let At,k(n)At,k(n) denote the number of partition k-tuples of n where each partition is t  -core. In this paper, we establish formulas of At,k(n)At,k(n) for some values of t and k   by employing the method of modular forms, which extends Wang's result for t=3t=3 and k=2,3k=2,3.