Generating series, also called generating functions, play an important role in combinatorial mathematics. Many enumeration problems can be solved by transferring the basic operations on sets into algebraic operations on formal series leading to a solution of an enumeration problem. The famous paper by Doubilet, Rota and Stanley, The idea of generating function [40], places the subject in a general mathematical frame-work allowing one to present in a unified way the diversity of generating functions, from the ordinary ones to the exponential or even Dirichlet.