A Grammatical Calculus for the Ramanujan Polynomials
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911672447270912 |
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| author | Chen, William Y. C. Fu, Amy M. Wang, Elena L. |
| author_facet | Chen, William Y. C. Fu, Amy M. Wang, Elena L. |
| contents | The Ramanujan polynomials arise in three intertwined contexts. As remarked by BerndtEvans-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 as a refinement of the classical formula of Cayley for 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01649 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Grammatical Calculus for the Ramanujan Polynomials Chen, William Y. C. Fu, Amy M. Wang, Elena L. Combinatorics 05A05, 05A15 The Ramanujan polynomials arise in three intertwined contexts. As remarked by BerndtEvans-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 as a refinement of the classical formula of Cayley for 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. |
| title | A Grammatical Calculus for the Ramanujan Polynomials |
| topic | Combinatorics 05A05, 05A15 |
| url | https://arxiv.org/abs/2506.01649 |