Séminaire Lotharingien de Combinatoire, B21m (1989), 13 pp.
[Formerly: Publ. I.R.M.A. Strasbourg, 1990, 413/S-21, p. 5-13.]

Pierre Bouchard and Mario Ouellette

Décomposition arborescente de Mario Ouellette

Abstract. Yeong-Nan Yeh proved the semiring (with respect to sum and product) of (isomorphism classes of) species to be factorial, more precisely isomorphic to the semi-ring of formal power series N[[M]] where M is the monoid (for .) of isomorphism classes of molecular species ([Yeh]). This amounts to saying that each species is uniquely a sum of products of atomic species. Studying also the behavior of the composition of species, Mario Ouellette ([Oue]) showed that each species has a unique decomposition as a composition of a primitive species and a molecular species: this leads to a unique "arborescent" decomposition for species. In this talk, we give a detailed demonstration of the lemma which is at the heart of his proof and a sketch his proof.

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