We give asurvey of basic tools and motivations in contemporary enumerative combinatorics. We start with the classical example of binary trees and generating function of Catalan numbers and extend it to “decomposable structures”. A typical but non trivial example is given by the Schaeffer decomposition of planar maps explaining the algebricity of the corresponding generating function. Then we systematically show the correspondence between algebraic operations about formal power series (generating function) and the corresponding operations at the level of combinatorial objects (sum, product, substitution, quasi-inverse, …). A section is given about rational generating functions and a basic inversion lemma (interpreted in physics as the transition matrix methodology). We also investigate the world of q-series and q-analogues, with some example of bijective proof identities and the “bijective paradigm”. We finish with the introduction of the theory of heaps of pieces and some inversion formula as a typical example of the active domain of algebraic combinatorics, in connection with theoretical physics.
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