Séminaire Lotharingien de Combinatoire, B14Si (1986), p.
60-70.
Gilbert Labelle
Interpolation dans les K-espèces
Abstract.
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).
The following version is available: