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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.23959 |
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| _version_ | 1866908995024846848 |
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| author | Han, Guo-Niu Ji, Kathy Q. Xiong, Huan |
| author_facet | Han, Guo-Niu Ji, Kathy Q. Xiong, Huan |
| contents | 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 $q$-André polynomials, including their generating functions and recurrences. This work extends the grammatical method to the $q$-setting and opens up new research directions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_23959 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $q$-Derivative Grammar Han, Guo-Niu Ji, Kathy Q. Xiong, Huan Combinatorics 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 $q$-André polynomials, including their generating functions and recurrences. This work extends the grammatical method to the $q$-setting and opens up new research directions. |
| title | $q$-Derivative Grammar |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.23959 |