Normal ordered grammars

TL;DR

This paper develops the theory of normal ordered grammars, providing combinatorial interpretations for Eulerian polynomials and related structures, revealing connections to Catalan numbers and type B Eulerian polynomials.

Contribution

It introduces the concept of normal ordered grammars and applies it to various Eulerian polynomial families, offering new grammatical interpretations and generating function insights.

Findings

Normal ordered grammars provide combinatorial interpretations for Eulerian polynomials.

The exponential generating function involves Catalan numbers as an exponent.

Normal ordered grammars relate to type B Eulerian polynomials and up-down run polynomials.

Abstract

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.