Combinatorial proofs of generating function identities for F-partitions

In his memoir in 1984, George E. Andrews introduces many general classes of Frobenius partitions (simply F-partitions). Especially, he focuses his interest on two general classes of F-partitions: one is F-partitions that allow up to k repetitions of an integer in any row, and the other is F-partitions whose parts are taken from k copies of the nonnegative integers. The latter are called k colored F-partitions or F-partitions with k colors. Andrews derives the generating functions of the number of F-partitions with k repetitions and F-partitions with k colors of n and leaves their purely combinatorial proofs as open problems. The primary goal of this article is to provide combinatorial proofs in answer to Andrews’ request.