Grammaires de Chen comme opérateurs différentiels sur les séries formelles

Grammars that are classically defined as sets of substitution rules over an alphabet have been recently regarded by William Chen as formal derivatives of polynomials and series in the elements of that alphabet.

In the first part of the lecture we give several classical examples of enumerative polynomials that can be generated by Chen grammars on finite alphabets. In the second part we consider the case of infinite alphabets: we recall the "Faa di Bruno grammar" given by Chen. Then we introduce two grammars respectively associated with the solutions of y'=f(y) and of y=f(xy), where f is an arbitrary formal series of exponential type. Those grammars have combinatorial interpretations, the latter one providing simple recurrences for the solution of the Lagrange inversion formula.