Résumé : Starting from a presentation of the many recent applications of Galois theory of functional equations to enumerative combinatorics, we will introduce the Galois theory of (different kinds) of difference equations. We will focus on the point of view of the applications, hence with little emphasis on the technicalities of the domain, but I'm willing to do an hour of « exercises » (i.e. to go a little deeper into the proofs), if a part of the audience is interested.

Keywords : D-finite; differential Galois theory; iterative equation; generating function; self-similar graphs; complete trees; pattern avoiding permutation; Combinatorial power series; differential transcendence; Sheffer sequences; Bell numbers; Bernoulli numbers; Euler numbers; Genocchi numbers