Polynomial species and connections among bases of the symmetric polynomials

In two papers published in 1968 G . C . Rota, carrying to the limit the algebraic processes which Baxter and others introduced for the resolution of problems generated by theory of probability, shows that every identity in a Baxter algebra is equivalent to an identity between symmetric functions. In his second paper the Author proves,through combinatorial methods, classical identities between symmetric functions which translate identities in Baxter algebras of probability interest. The concept of generating function of a function set, conveyed in these papers, is developed later by Doubilet, Rota and Stanley (see [ 4 ] , [9]).These Authors introduce a process for the construction of algebras of generating functions, both classical and innovative, for the resolution of enumerative problems. Using the concept of generating function of a function set and techniques involving the lattice o f partition of a set, Doubilet (see [ 3 ] ) derives many of the known results and new ones about symmetric functions. In many cases Doubilet utilizes the Mobius inversion formula, but he also succeds in giving bijective proofs of identities between symmetric functions. These proofs consist essentially in an interpretation o f the functions that occur in the identities in terms of sets and in finding a bijections between them. In this paper we use the theory of polynomial species (see [ l ] ) which gives a systematic approach to this kind o f proof. We prove with bijective arguments, some identities which occur among the classical bases of symmetric polynomials of degree n. The language is that of categories theory and this emphasizes the generality degree of the concept of species.