Congruences involving generalized Frobenius partitions (Q1210476)

The main result of this paper is a congruence involving the function \(\overline{c\varphi_m}(n)\) which denotes the number of generalized Frobenius partitions of \(n\) with \(m\) colors whose order is \(m\) under cyclic permutations of the \(m\) colors. L. W. Kolitsch has proved that for \(m\geq 2\) and all \(n\geq 1\), \(\overline{c\varphi_ m}(n)\equiv 0\pmod{m^2}\). The following is proved: For \(m=5, 7\) and \(11\) and for all \(n\geq 1\), \(\overline{c\varphi_ m}(n) \equiv 0\pmod{m^3}\).