Interpolation dans les K-espèces

Gilbert Labelle

Séminaire lotharingien de combinatoire, 14s (1986)

Résumé

Let F be a species without structure on the empty set and wwith only one singleton (e.g., the speciees of circular permutations), and let H be an arbitrary species. For n ∈ N we write F^<n> = F o ... o F for the (n-fold iterated) species of F-arborescences of height ≤ n, and we write H o F^<n> for the species of H-forests of such arborescences. It is our aim to give a combinatorial meaning to H o F^<t> for values of t that are not positive integers. We show that if t ∈ K, where K is a binomial ring, then H o F^<t> is a K-speecies in the sense of Y.N. Yeh. The results also holds if H and F ar themselves K-species, as well as in the multisort case. Our approach is simple: it consists of an adaptation, in the context of K-species, of the classical interpolation formula of Newton. This approach has already been used by the author to implement a "continuous" iteration of formal power series, and by Joyal for combinatorially implementing the inverse (t = -1) of virual species (i.e., of Z-species).