Normal ordering problem and the extensions of the Stirling grammar

Abstract: The purpose of this paper is to investigate the connection between context-free grammars and normal ordering problem, and then to explore various extensions of the Stirling grammar. We present grammatical characterizations of several well known combinatorial sequences, including the generalized Stirling numbers of the second kind related to the normal ordering problem and the $r$-Dowling polynomials. Also, possible avenues for future research are described.