$q$-Derivative Grammar
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 -analogue of context-free grammars, which we call the -derivative grammar. We establish the basic framework of -grammars and develop the -grammar calculus for computing -exponential generating functions associated with -grammars. Concrete -grammars are constructed to study -Eulerian, -Roselle and -Andr\'e polynomials, including their generating functions and recurrences. This work extends the grammatical method to the -setting and opens up new research directions.
Cite
@article{arxiv.2604.23959,
title = {$q$-Derivative Grammar},
author = {Guo-Niu Han and Kathy Q. Ji and Huan Xiong},
journal= {arXiv preprint arXiv:2604.23959},
year = {2026}
}
Comments
66 pages, 8 figures