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.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics