标题： 拉马努金多项式的语法演算 显示英文标题

标题： A Grammatical Calculus for the Ramanujan Polynomials

摘要： 正如Berndt所指出的，在拉马努金多项式的原始定义中似乎没有提到组合学的观点。在另一个场景中，Shor和Dumont-Ramamonjisoa独立地设计了一种生成根树的递归算法。Zeng发现了拉马努金多项式与按不当边数量枚举根树之间的联系。我们通过引入额外标签，提出了一种根树的适当标记方案。借助这一语法，我们发展了一种严重依赖于拉马努金多项式常数性质的演算方法。从语法公式出发，我们重新得到了拉马努金关于隐函数的定义方程。因此，拉马努金的两个主题汇聚到一个组合结构上。此外，我们提供了对Shor和Berndt-Evans-Wilson独立提出的递归背后双射关系的一种语法处理方法。 显示英文摘要

摘要： As remarked by Berndt, no combinatorial perspective seems to be alluded in the original definition of the Ramanujan polynomials. On a different scene, a recursive algorithm to generate rooted trees has been devised independently by Shor and Dumont-Ramamonjisoa. Zeng discovered the connection between the Ramanujan polynomials and the enumeration of rooted trees by number of improper edges. We present a proper labeling scheme for rooted trees by employing an extra label. Harnessed by this grammar, we develop a calculus heavily depending on the constant properties for the Ramanujan polynomials. From the grammatical formulation, we recover the defining equation of Ramanujan on an implicit function. So the two themes of Ramanujan converge to one combinatorial structure. Moreover, we provide a grammatical treatment of a bijection behind the recursion independently due to Shor and Berndt-Evans-Wilson.