Finite Interpretation of the Hyper-Catalan Series Zero and its Powers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912529878351872 |
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| author | Rubine, Dean Mukewar, Pratham |
| author_facet | Rubine, Dean Mukewar, Pratham |
| contents | In 2025, Wildberger and Rubine showed the formal series zero of the univariate geometric polynomial is $\mathbf{S}$, the generating series for the hyper-Catalan numbers $\mathbf{C}_m$, which count the number of roofed subdivided polygons (subdigons) of type $\mathbf{m}$. We show that we can interpret this result as a finite identity at each level, where a level is a truncation of $\Sb$ to a given maximum number of vertices, edges, or faces (bounded by degree) of the associated subdigon types. We then explore powers $\mathbf{S}^r$, recounting Raney's and our own combinatorial derivations of its coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06739 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite Interpretation of the Hyper-Catalan Series Zero and its Powers Rubine, Dean Mukewar, Pratham Combinatorics In 2025, Wildberger and Rubine showed the formal series zero of the univariate geometric polynomial is $\mathbf{S}$, the generating series for the hyper-Catalan numbers $\mathbf{C}_m$, which count the number of roofed subdivided polygons (subdigons) of type $\mathbf{m}$. We show that we can interpret this result as a finite identity at each level, where a level is a truncation of $\Sb$ to a given maximum number of vertices, edges, or faces (bounded by degree) of the associated subdigon types. We then explore powers $\mathbf{S}^r$, recounting Raney's and our own combinatorial derivations of its coefficients. |
| title | Finite Interpretation of the Hyper-Catalan Series Zero and its Powers |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2508.06739 |