Bijective proofs of certain vector partition identities

Abstract

for 1 <; i < k. L. Carlitz [2] first derived the generating function for restricted bipartite partitions. Subsequently Carlitz and Roselle [3] enumerated certain special families of these partitions e.g., restricted bipartite partitions where the m< and % are all odd. Finally both Roselle [4] and Andrews [1] have obtained different generalizations for multipartite partitions. All these results have been proved by manipulation of formal power series. However, the simplicity of the generating functions obtained suggests that purely combinatorial methods could be applied. The purpose of this paper is to give direct bijective proofs which, in addition, permit us to count a new family of partitions.