#####
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.