In (Bessenrodt, 1991) a combinatorial proof of a refinement of the Andrews-Olsson partition identity (Andrews and Olsson, 1991) was given. In this article we explore the methods of (Bessenrodt, 1991) further to deal also with partitions where repetitions are allowed and where the restrictions on the congruence set are omitted. We obtain a considerable generalisation of the earlier results, again giving bijective proofs of the partition identities in question. As an application, it is then shown how to deduce (refinements of) some old results by Sylvester and Schur as well as some recent results by Alladi and Gordon [1].