  We introduce the theory of normal ordered grammars, which gives a natural
generalization of the normal ordering problem. To illustrate the main idea, we
explore normal ordered grammars associated with the Eulerian polynomials and
the second-order Eulerian polynomials. In particular, we present a normal
ordered grammatical interpretation for the (cdes,cyc) (p,q)-Eulerian
polynomials, where cdes and cyc are the cycle descent and cycle statistics,
respectively. The exponential generating function for a family of polynomials,
generated by a normal ordered grammar associated with the second-order Eulerian
polynomials, reveals an interesting feature: its expression involves the
generating function for Catalan numbers as its exponent. In the final part, we
discuss some normal ordered grammars related to the type B Eulerian
polynomials. A normal ordered grammatical interpretation of the up-down run
polynomial is also established.
