Symmetric functions and bijective identities

One of the latest interests of Combinatorics is the development of a systematic theory of bijective proofs. In this context the idea of proving identities among symmetric functions by bijective arguments comes in again. The first attempt of a systematic approach to this point of view is thanks to G.C. Rota ( 3 1 , resumed later by P. Doubilet ( 2 ) and developed by F.Bonetti, G.C. Rota, D. Senato and A. Venezia in a paper which is yet to be published (1). In (1) two complementary notions are introduced: Formal Polynomials and Polynomial Species that are the key of a categorical setting for bijective proofs of identities among symmetric functions. In this paper a more general notion of Polynomial Species is introduced, which permits us to extend the techniques which we have used in (1) to the symmetric functions in two sets of indipendent variables.