{"paper":{"title":"Ordered trees and the Geode","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fern Gossow","submitted_at":"2025-07-24T05:18:18Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.18097","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.18097/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}