bibkey: han2026qgrammar authors: Guo-Niu Han, Kathy Q. Ji, Huan Xiong year: 2026 title: “q-Derivative Grammar” doi: 10.48550/arXiv.2604.23959 url: https://arxiv.org/abs/2604.23959v2 claim: “A formal q-analogue of Chen’s context-free grammars with rules, orders and a q-derivative operator; applications to q-tangent and q-secant numbers, and an appendix of conjectured counts of distinct terms such as Conjecture III.7 for the grammar G_Sec.” strata_touched:

• D5/S3/Combinatorics/QGrammar/SecGrammar license: citation-only triage: anchor

Han, Ji and Xiong, q-derivative grammar

A q-derivative grammar consists of a set of indexed variables, a rule assigning to each variable an expression, and an order that rearranges the variables of a word. The q-derivative D replaces one variable of a word by its rule, raises the indices of the variables to its right by one, and rearranges the result by the order; it is extended linearly to formal sums of words with coefficients in ℤ[q]. Appendix III records counts of distinct terms of Dⁿ for several grammars. For the grammar G_Sec with rules x_j ↦ q^j(1 + x_j x_{j+1}), y_j ↦ q^j y_j x_{j+1} and the descending interleaving order, Conjecture III.7 gives the number of distinct terms of Dⁿ(y₀) as an explicit cubic quasipolynomial in n.

The module D5/S3/Combinatorics/QGrammar/SecGrammar proves Conjecture III.7.

Verified locator

DOI: 10.48550/arXiv.2604.23959

URL: https://arxiv.org/abs/2604.23959v2

• Locator: Section 3, Definitions 3.3–3.7.
• Locator: Appendix III, the grammar G_Sec and Conjecture III.7.