AI 中文总结
介绍源自罗塔符号演算的语法演算，利用微分算子性质进行演算以推导组合恒等式和计算生成函数，语法标记是桥梁，语法可辅助构建双射，许多组合结构有上下文无关递归构造。
AI 中文摘要
我们介绍了源自罗塔的符号演算的语法演算。上下文无关语法出现在计算机科学中。微分算子可被视为上下文无关语法的产生规则。借助微分算子的性质，我们可以在严格的基础上进行语法演算，以推导组合恒等式和计算生成函数。语法标记是组合结构与语法之间的桥梁。语法有助于在具有相同语法的两个组合结构之间构建双射，这种双射称为语法辅助双射。许多组合结构都有某种意义上上下文无关的递归构造，比如与欧拉多项式、贝尔多项式、拉马努金多项式和纳拉亚纳多项式对应的那些。
英文摘要
We give an introduction to the grammatical calculus, which originated from the umbral calculus of Rota. Context-free grammars arise in computer science. Differential operators can be viewed as the production rules of context-free grammars. By the properties of differential operators, we may perform the grammatical calculus on a rigorous basis, for the purposes of deriving combinatorial identities and computing generating functions. A grammatical labeling serves as a bridge between a combinatorial structure and the grammar. A grammar may be instrumental in constructing a bijection between two combinatorial structures sharing the same grammar. Such a bijection is called a grammar assisted bijection. Many combinatorial structures admit recursive constructions that are context-free in some sense, such as those corresponding to the Eulerian polynomials, Bell polynomials, the Ramanujan polynomials and the Narayana polynomials.
Comments54 pages, 6 figures, English Translation, The Grammatical Calculus (in Chinese), Scientia Sinica Mathematica 56(9) (2026) 1-34
