Symmetric functions and symmetric species

The idea of proving identities for symmetric functions by bijective arguments is quite old; it goes back to Lucas (Theorie des nombres, 1891) and probably earlier. To the best of our knowledge, the first glimmerings of a systematization of such bijective arguments goes back to one of the present author (cf. 191); the idea was further developed by R.P. Stanley, wo gave a bijective proof of Waring’s formula by Mobius inversion on the lattice of partition of a set, and later by Doubilet, who gaves bijective proofs of several identities in the theory of symmetric functions. Joyal’s theory of species led us t o develop a systematic setting for such bijective proofs. We introduce here the notion of synirnetric species, which can be viewed as a set-theoretic (a category-theoretic) counterpart of the notion of a symmetric function. To each of the classical classes of symmetric functions we associate a symmetric species. Operations on species, as introduced by Joyal, are generalized to symmetric species, and simple categorical operations yielded bijective proofs of all identities among elementary symmetric functions. By way of example, we give a bijective proof of Waring’s forniula, which we believe to be new, and dispenses