{"paper":{"title":"Lattice paths and the Geode","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ira M. Gessel","submitted_at":"2025-07-12T21:16:49Z","abstract_excerpt":"Let $t_1,t_2,\\dots$ be variables, and let $S$ be the formal power series in the variables $t_1, t_2,\\dots$ satisfying $S=1+\\sum_{i=1}^\\infty t_n S^n.$ Let $S_1 =\\sum_{n=1}^\\infty t_n$. Wildberger and Rubine recently showed that there is a formal power series $G$ in the $t_i$, which they called the Geode, satisfying $S=1+GS_1$. In this paper we discuss some of the properties of the Geode and of the related series $H=G/S$, which satisfies $S=1/(1-HS_1)$. We show that \\begin{equation*} G=\\biggl(1-\\sum_{n=1}^\\infty t_n (1+S+S^2+\\cdots+S^{n-1})\\biggr)^{-1}, \\end{equation*} and \\begin{equation*} H=\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.09405","kind":"arxiv","version":2},"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/2507.09405/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"}