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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=\\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2507.09405","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-07-12T21:16:49Z","cross_cats_sorted":[],"title_canon_sha256":"22d26e6e191d7ed1813b3c3e6553eae52dac6789179a73d110671d280f75ba51","abstract_canon_sha256":"0e5ec30f85ab71760d7a9a24554821f4091d5b16584c6987375553b6476d2633"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:39:13.468968Z","signature_b64":"nuFkPHHy57QpYfm0EUvRZBHuInOOsYSuOsVYFRMFvJThSMzX29aSxwBRmgcRWq5eC+NDuo3fBHmOgn75V2yfCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d20ba47057e75b4f31111b51ac639349d235d2895e704e7967ad6cade00f001","last_reissued_at":"2026-07-05T11:39:13.468563Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:39:13.468563Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2507.09405","created_at":"2026-07-05T11:39:13.468617+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.09405v2","created_at":"2026-07-05T11:39:13.468617+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.09405","created_at":"2026-07-05T11:39:13.468617+00:00"},{"alias_kind":"pith_short_12","alias_value":"TUQLURYFPZ23","created_at":"2026-07-05T11:39:13.468617+00:00"},{"alias_kind":"pith_short_16","alias_value":"TUQLURYFPZ23J4YR","created_at":"2026-07-05T11:39:13.468617+00:00"},{"alias_kind":"pith_short_8","alias_value":"TUQLURYF","created_at":"2026-07-05T11:39:13.468617+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2507.18097","citing_title":"Ordered trees and the Geode","ref_index":3,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TUQLURYFPZ23J4YRCG2RVRRZGS","json":"https://pith.science/pith/TUQLURYFPZ23J4YRCG2RVRRZGS.json","graph_json":"https://pith.science/api/pith-number/TUQLURYFPZ23J4YRCG2RVRRZGS/graph.json","events_json":"https://pith.science/api/pith-number/TUQLURYFPZ23J4YRCG2RVRRZGS/events.json","paper":"https://pith.science/paper/TUQLURYF"},"agent_actions":{"view_html":"https://pith.science/pith/TUQLURYFPZ23J4YRCG2RVRRZGS","download_json":"https://pith.science/pith/TUQLURYFPZ23J4YRCG2RVRRZGS.json","view_paper":"https://pith.science/paper/TUQLURYF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.09405&json=true","fetch_graph":"https://pith.science/api/pith-number/TUQLURYFPZ23J4YRCG2RVRRZGS/graph.json","fetch_events":"https://pith.science/api/pith-number/TUQLURYFPZ23J4YRCG2RVRRZGS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TUQLURYFPZ23J4YRCG2RVRRZGS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TUQLURYFPZ23J4YRCG2RVRRZGS/action/storage_attestation","attest_author":"https://pith.science/pith/TUQLURYFPZ23J4YRCG2RVRRZGS/action/author_attestation","sign_citation":"https://pith.science/pith/TUQLURYFPZ23J4YRCG2RVRRZGS/action/citation_signature","submit_replication":"https://pith.science/pith/TUQLURYFPZ23J4YRCG2RVRRZGS/action/replication_record"}},"created_at":"2026-07-05T11:39:13.468617+00:00","updated_at":"2026-07-05T11:39:13.468617+00:00"}