The Grammatical Calculus

We give an introduction to the grammatical calculus, which originated from the umbral calculus of Rota. Context-free grammars arise in computer science. Differential operators can be viewed as the production rules of context-free grammars. By the properties of differential operators, we may perform the grammatical calculus on a rigorous basis, for the purposes of deriving combinatorial identities and computing generating functions. A grammatical labeling serves as a bridge between a combinatorial structure and the grammar. A grammar may be instrumental in constructing a bijection between two combinatorial structures sharing the same grammar. Such a bijection is called a grammar assisted bijection. Many combinatorial structures admit recursive constructions that are context-free in some sense, such as those corresponding to the Eulerian polynomials, Bell polynomials, the Ramanujan polynomials and the Narayana polynomials.