A simple proof of some congruences for colored generalized frobenius partitions

where c#,Jr) is the number of F-partitions of r using h colors with (at most) s repetitions where s can be any positive integer or 00 (to represent no restriction on repetitions). The proofs of these congruences were based on some interesting congruence properties of compositions and were combinatorial in nature. Though the proofs were straightforward, they were somewhat lengthy and tedious. During some recent work with colored F-partitions the following analytic proof of the above congruences was discovered. This alternate proof highlights yet another combinatorial property of these colored F-partitions. As was shown in each of the papers [2,3] the sum,