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<p>The Ramanujan polynomials arise in three intertwined contexts. As remarked by Berndt-Evans-Wilson, no combinatorial perspective seems to be alluded to in the original definition of Ramanujan. On a different stage, Dumont-Ramamonjisoa uncovered a combinatorial structure underneath an equation also considered by Ramanujan. Around the same time, Shor came up with the same construction yielding a refinement of the classical formula of Cayley for labeled trees. We present a labeling scheme for rooted trees by employing an extra label marking improper edges. Harnessed by this grammar, we develop a grammatical calculus for the Ramanujan polynomials heavily relying on the constant properties. Moreover, we provide a grammatical formulation of a correspondence that leads to the recurrence relation due to Berndt-Evans-Wilson and Shor.</p>

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A grammatical calculus for the Ramanujan polynomials

  • 期刊论文

  • 发表日期：

    2026年6月1日

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  • William Y. C. Chen,
  • Amy M. Fu,
  • Elena L. Wang

摘要

The Ramanujan polynomials arise in three intertwined contexts. As remarked by Berndt-Evans-Wilson, no combinatorial perspective seems to be alluded to in the original definition of Ramanujan. On a different stage, Dumont-Ramamonjisoa uncovered a combinatorial structure underneath an equation also considered by Ramanujan. Around the same time, Shor came up with the same construction yielding a refinement of the classical formula of Cayley for labeled trees. We present a labeling scheme for rooted trees by employing an extra label marking improper edges. Harnessed by this grammar, we develop a grammatical calculus for the Ramanujan polynomials heavily relying on the constant properties. Moreover, we provide a grammatical formulation of a correspondence that leads to the recurrence relation due to Berndt-Evans-Wilson and Shor.