{"paper":{"title":"Hyper-Catalan and Geode Recurrences and Three Conjectures of Wildberger","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dean Rubine","submitted_at":"2025-07-06T22:03:56Z","abstract_excerpt":"The hyper-Catalan number $C[m_2,m_3,m_4,\\ldots]$ counts the number of subdivisions of a roofed polygon into $m_2$ triangles, $m_3$ quadrilaterals, $m_4$ pentagons, etc. Its closed form has been known since Erd\\'elyi and Etherington, 1940. In 2025, Wildberger and Rubine showed its generating sum $\\mathbf{S}[t_2,t_3,t_4,\\ldots]$ is a zero of the general geometric univariate polynomial. We use that to derive a recurrence for hyper-Catalans, which expresses each in terms of other hyper-Catalans with smaller indices, generalizing the well-known Catalan convolution sum.\n  Wildberger notes the factor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.04552","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/2507.04552/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"}