Séminaire Lotharingien de Combinatoire, B17b (1987).
[Formerly: Publ. I.R.M.A. Strasbourg, 1987, 348/S-17, p.
103-127.]
Gilbert Labelle
Dérivées directionnelles et
développements de Taylor combinatoires
Abstract.
Let A be the set of all isomorphism classes of atomic species,
let K be a binomial half-ring and K-
its rational closure. The differential half-ring
K[[A]] of all K-species in the sense of
Yeh is a combinatorial and algebraic extension of the
half-ring K[[X]] of all formal power series
in one indeterminate X. Using the operation of
substitution in K[[A]] and the
Q-species X^ of "pseudo-singletons"
we study two new notions: the combinatorial directional
derivative of a K-species in the direction of
another K-species and Taylor expansions in
K[[A]]. The use of K--species
is essential here. We show, along the way, certain
similarities and differences between these new notions and
their classical analogues in K[[X]].
Tables are given for small cardinalities.
The paper has been finally published under the same title in
Discrete Math. 79 (1990), 279-297.