Ordered trees and the Geode
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911074320646144 |
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| author | Gossow, Fern |
| author_facet | Gossow, Fern |
| contents | In recent work of Wildberger and Rubine, it is shown that the formal power series $\mathbf{S}$ in the variables $t_1,t_2,\dots$ satisfying $\mathbf{S}=1+\sum_{n\geq 1} t_n\mathbf{S}^n$ has a factorisation $\mathbf{S}=1+(t_1+t_2+\cdots)\mathbf{G}$, where $\mathbf{G}$ is a power series with nonnegative coefficients called the Geode. In this note we give a combinatorial interpretation for the coefficients of $\mathbf{G}$ based on ordered trees. This amends the statement of a disproved conjecture of Wildberger and Rubine which suggests a similar (but incorrect) interpretation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_18097 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ordered trees and the Geode Gossow, Fern Combinatorics 05A19, 05C05 In recent work of Wildberger and Rubine, it is shown that the formal power series $\mathbf{S}$ in the variables $t_1,t_2,\dots$ satisfying $\mathbf{S}=1+\sum_{n\geq 1} t_n\mathbf{S}^n$ has a factorisation $\mathbf{S}=1+(t_1+t_2+\cdots)\mathbf{G}$, where $\mathbf{G}$ is a power series with nonnegative coefficients called the Geode. In this note we give a combinatorial interpretation for the coefficients of $\mathbf{G}$ based on ordered trees. This amends the statement of a disproved conjecture of Wildberger and Rubine which suggests a similar (but incorrect) interpretation. |
| title | Ordered trees and the Geode |
| topic | Combinatorics 05A19, 05C05 |
| url | https://arxiv.org/abs/2507.18097 |