{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:WLZH5NPUPOLNKEVDFU35B2DPRE","short_pith_number":"pith:WLZH5NPU","schema_version":"1.0","canonical_sha256":"b2f27eb5f47b96d512a32d37d0e86f892e0a2d03397f25b99a5be704f0e37156","source":{"kind":"arxiv","id":"2607.27397","version":1},"attestation_state":"computed","paper":{"title":"Self-avoiding polygons on a three-row square-lattice strip","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.CO","authors_text":"Michael von Thaden","submitted_at":"2026-07-29T19:10:42Z","abstract_excerpt":"We give a closed formula for the number $p^{S_2}(n)$ of self-avoiding polygons (SAPs) of length $n$ on the strip $S_2:=\\mathbb{Z}\\times\\{0,1,2\\}$, together with closed formulas for those subtypes of SAPs which are determined by the numbers of vertical steps in their leftmost and rightmost columns. For the subtype whose leftmost and rightmost columns each contain two vertical steps, we also derive an alternative representation as a binomial sum. Our derivation is elementary: it is purely combinatorial and geometric and avoids generating functions. Comparing the two representations yields a new "},"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":"2607.27397","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-29T19:10:42Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"47fb48d30160c1df5716f3956d025641abb8bb2885842bc007127dd3e2eac7a9","abstract_canon_sha256":"3e1723a2a43f0bcced401c662f243b145b62c5189a0a8c177e5a23279cf4a3cb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b2f27eb5f47b96d512a32d37d0e86f892e0a2d03397f25b99a5be704f0e37156","last_reissued_at":"2026-07-31T00:11:40.689675Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-31T00:11:40.689675Z"},"graph_snapshot":{"paper":{"title":"Self-avoiding polygons on a three-row square-lattice strip","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.CO","authors_text":"Michael von Thaden","submitted_at":"2026-07-29T19:10:42Z","abstract_excerpt":"We give a closed formula for the number $p^{S_2}(n)$ of self-avoiding polygons (SAPs) of length $n$ on the strip $S_2:=\\mathbb{Z}\\times\\{0,1,2\\}$, together with closed formulas for those subtypes of SAPs which are determined by the numbers of vertical steps in their leftmost and rightmost columns. For the subtype whose leftmost and rightmost columns each contain two vertical steps, we also derive an alternative representation as a binomial sum. Our derivation is elementary: it is purely combinatorial and geometric and avoids generating functions. Comparing the two representations yields a new "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.27397","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/2607.27397/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":"2607.27397","created_at":"2026-07-31T00:11:40.691991+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.27397v1","created_at":"2026-07-31T00:11:40.691991+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.27397","created_at":"2026-07-31T00:11:40.691991+00:00"},{"alias_kind":"pith_short_12","alias_value":"WLZH5NPUPOLN","created_at":"2026-07-31T00:11:40.691991+00:00"},{"alias_kind":"pith_short_16","alias_value":"WLZH5NPUPOLNKEVD","created_at":"2026-07-31T00:11:40.691991+00:00"},{"alias_kind":"pith_short_8","alias_value":"WLZH5NPU","created_at":"2026-07-31T00:11:40.691991+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/WLZH5NPUPOLNKEVDFU35B2DPRE","json":"https://pith.science/pith/WLZH5NPUPOLNKEVDFU35B2DPRE.json","graph_json":"https://pith.science/api/pith-number/WLZH5NPUPOLNKEVDFU35B2DPRE/graph.json","events_json":"https://pith.science/api/pith-number/WLZH5NPUPOLNKEVDFU35B2DPRE/events.json","paper":"https://pith.science/paper/WLZH5NPU"},"agent_actions":{"view_html":"https://pith.science/pith/WLZH5NPUPOLNKEVDFU35B2DPRE","download_json":"https://pith.science/pith/WLZH5NPUPOLNKEVDFU35B2DPRE.json","view_paper":"https://pith.science/paper/WLZH5NPU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.27397&json=true","fetch_graph":"https://pith.science/api/pith-number/WLZH5NPUPOLNKEVDFU35B2DPRE/graph.json","fetch_events":"https://pith.science/api/pith-number/WLZH5NPUPOLNKEVDFU35B2DPRE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WLZH5NPUPOLNKEVDFU35B2DPRE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WLZH5NPUPOLNKEVDFU35B2DPRE/action/storage_attestation","attest_author":"https://pith.science/pith/WLZH5NPUPOLNKEVDFU35B2DPRE/action/author_attestation","sign_citation":"https://pith.science/pith/WLZH5NPUPOLNKEVDFU35B2DPRE/action/citation_signature","submit_replication":"https://pith.science/pith/WLZH5NPUPOLNKEVDFU35B2DPRE/action/replication_record"}},"created_at":"2026-07-31T00:11:40.691991+00:00","updated_at":"2026-07-31T00:11:40.691991+00:00"}