$q$-Derivative Grammar

TL;DR

This paper introduces the $q$-derivative grammar, extending context-free grammars to the $q$-setting, and develops calculus for $q$-generating functions, applied to various $q$-polynomials.

Contribution

It establishes the framework of $q$-grammars and develops $q$-calculus, enabling analysis of $q$-polynomials and extending grammatical methods in combinatorics.

Findings

Constructed concrete $q$-grammars for $q$-Eulerian, $q$-Roselle, and $q$-André polynomials.

Derived generating functions and recurrences for these $q$-polynomials.

Extended grammatical method to the $q$-setting, opening new research directions.

Abstract

The concept of context-free grammar in Combinatorics was first introduced by Chen in 1993. In 1996, Dumont significantly extended the theory of context-free grammars to a variety of other combinatorial models. Substantial progress in this direction has been achieved over the last decade. In this paper, we introduce a $q$-analogue of context-free grammars, which we call the $q$-derivative grammar. We establish the basic framework of $q$-grammars and develop the $q$-grammar calculus for computing $q$-exponential generating functions associated with $q$-grammars. Concrete $q$-grammars are constructed to study $q$-Eulerian, $q$-Roselle and $q$-Andr\'e polynomials, including their generating functions and recurrences. This work extends the grammatical method to the $q$-setting and opens up new research directions.