{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:TOT2XHWXRD22AARWELUIBCENSZ","short_pith_number":"pith:TOT2XHWX","schema_version":"1.0","canonical_sha256":"9ba7ab9ed788f5a0023622e880888d965627ff5978a3ada660c3b9af7c0d75c4","source":{"kind":"arxiv","id":"1512.00080","version":2},"attestation_state":"computed","paper":{"title":"A new shellability proof of an identity of Dixon","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Augustine O'Keefe, Daniel Parry, Ruth Davidson","submitted_at":"2015-11-30T22:51:57Z","abstract_excerpt":"We give a new proof of an old identity of Dixon (1865-1936) that uses tools from topological combinatorics. Dixon's identity is re-established by constructing an infinite family of non-pure simplicial complexes $\\Delta(n)$, indexed by the positive integers, such that the alternating sum of the numbers of faces of $\\Delta(n)$ of each dimension is the left-hand side of the identity. We show that $\\Delta(n)$ is shellable for all $n$. Then, using the fact that a shellable simplicial complex is homotopy equivalent to a wedge of spheres, we compute the Betti numbers of $\\Delta(n)$ by counting (via a"},"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":"1512.00080","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-11-30T22:51:57Z","cross_cats_sorted":[],"title_canon_sha256":"21d8b4e84c9b9545e447959ff1f309c1c38d6049864f755392f789b5d5f987a6","abstract_canon_sha256":"cbaa3b9ebfacc8d55c8ebf5e7019b2a0681f9f919af1b9429440c9dbb6de91cb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:06.496328Z","signature_b64":"ueAegbigc7MLNFiklX938YWxcuzdoIPqkem2LI2NF/IVt5L8fyuq4Trm2npDumRC+Fj7yfoer/04gLtwQ7R8Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9ba7ab9ed788f5a0023622e880888d965627ff5978a3ada660c3b9af7c0d75c4","last_reissued_at":"2026-05-18T01:15:06.495825Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:06.495825Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A new shellability proof of an identity of Dixon","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Augustine O'Keefe, Daniel Parry, Ruth Davidson","submitted_at":"2015-11-30T22:51:57Z","abstract_excerpt":"We give a new proof of an old identity of Dixon (1865-1936) that uses tools from topological combinatorics. Dixon's identity is re-established by constructing an infinite family of non-pure simplicial complexes $\\Delta(n)$, indexed by the positive integers, such that the alternating sum of the numbers of faces of $\\Delta(n)$ of each dimension is the left-hand side of the identity. We show that $\\Delta(n)$ is shellable for all $n$. Then, using the fact that a shellable simplicial complex is homotopy equivalent to a wedge of spheres, we compute the Betti numbers of $\\Delta(n)$ by counting (via a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.00080","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":""},"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":"1512.00080","created_at":"2026-05-18T01:15:06.495904+00:00"},{"alias_kind":"arxiv_version","alias_value":"1512.00080v2","created_at":"2026-05-18T01:15:06.495904+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1512.00080","created_at":"2026-05-18T01:15:06.495904+00:00"},{"alias_kind":"pith_short_12","alias_value":"TOT2XHWXRD22","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"TOT2XHWXRD22AARW","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"TOT2XHWX","created_at":"2026-05-18T12:29:42.218222+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/TOT2XHWXRD22AARWELUIBCENSZ","json":"https://pith.science/pith/TOT2XHWXRD22AARWELUIBCENSZ.json","graph_json":"https://pith.science/api/pith-number/TOT2XHWXRD22AARWELUIBCENSZ/graph.json","events_json":"https://pith.science/api/pith-number/TOT2XHWXRD22AARWELUIBCENSZ/events.json","paper":"https://pith.science/paper/TOT2XHWX"},"agent_actions":{"view_html":"https://pith.science/pith/TOT2XHWXRD22AARWELUIBCENSZ","download_json":"https://pith.science/pith/TOT2XHWXRD22AARWELUIBCENSZ.json","view_paper":"https://pith.science/paper/TOT2XHWX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1512.00080&json=true","fetch_graph":"https://pith.science/api/pith-number/TOT2XHWXRD22AARWELUIBCENSZ/graph.json","fetch_events":"https://pith.science/api/pith-number/TOT2XHWXRD22AARWELUIBCENSZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TOT2XHWXRD22AARWELUIBCENSZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TOT2XHWXRD22AARWELUIBCENSZ/action/storage_attestation","attest_author":"https://pith.science/pith/TOT2XHWXRD22AARWELUIBCENSZ/action/author_attestation","sign_citation":"https://pith.science/pith/TOT2XHWXRD22AARWELUIBCENSZ/action/citation_signature","submit_replication":"https://pith.science/pith/TOT2XHWXRD22AARWELUIBCENSZ/action/replication_record"}},"created_at":"2026-05-18T01:15:06.495904+00:00","updated_at":"2026-05-18T01:15:06.495904+00:00"}