{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:TUYYHAHSIRD52LXTRKWKSTMFMS","short_pith_number":"pith:TUYYHAHS","schema_version":"1.0","canonical_sha256":"9d318380f24447dd2ef38aaca94d856482374b94de539adaef9ff6ca4fa81c9f","source":{"kind":"arxiv","id":"0905.4751","version":1},"attestation_state":"computed","paper":{"title":"Updown numbers and the initial monomials of the slope variety","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Jennifer D. Wagner, Jeremy L. Martin","submitted_at":"2009-05-28T21:09:52Z","abstract_excerpt":"Let $I_n$ be the ideal of all algebraic relations on the slopes of the $\\binom{n}{2}$ lines formed by placing $n$ points in a plane and connecting each pair of points with a line. Under each of two natural term orders, the initial ideal of $I_n$ is generated by monomials corresponding to permutations satisfying a certain pattern-avoidance condition. We show bijectively that these permutations are enumerated by the updown (or Euler) numbers, thereby obtaining a formula for the number of generators of the initial ideal of $I_n$ in each degree."},"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":"0905.4751","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-05-28T21:09:52Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"871f01f111e49dbd6d2d05c2a46f5894492f141fdd6f0a2db46349f16005d5f6","abstract_canon_sha256":"78353c34526d9d98152e7ae8d0b85d0e6982ab4ce3007a90849d560720bc8845"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:11:49.345529Z","signature_b64":"d752Jubipqyt0HuqrxFhU04LLOlSwjn7+nArbQeNtlEnnOszNHtOaSf7MqEQcMpcNU6OEq2g/nX6yk8h8Q+3Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d318380f24447dd2ef38aaca94d856482374b94de539adaef9ff6ca4fa81c9f","last_reissued_at":"2026-05-18T04:11:49.344807Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:11:49.344807Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Updown numbers and the initial monomials of the slope variety","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.CO","authors_text":"Jennifer D. Wagner, Jeremy L. Martin","submitted_at":"2009-05-28T21:09:52Z","abstract_excerpt":"Let $I_n$ be the ideal of all algebraic relations on the slopes of the $\\binom{n}{2}$ lines formed by placing $n$ points in a plane and connecting each pair of points with a line. Under each of two natural term orders, the initial ideal of $I_n$ is generated by monomials corresponding to permutations satisfying a certain pattern-avoidance condition. We show bijectively that these permutations are enumerated by the updown (or Euler) numbers, thereby obtaining a formula for the number of generators of the initial ideal of $I_n$ in each degree."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0905.4751","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":""},"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":"0905.4751","created_at":"2026-05-18T04:11:49.344931+00:00"},{"alias_kind":"arxiv_version","alias_value":"0905.4751v1","created_at":"2026-05-18T04:11:49.344931+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0905.4751","created_at":"2026-05-18T04:11:49.344931+00:00"},{"alias_kind":"pith_short_12","alias_value":"TUYYHAHSIRD5","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_16","alias_value":"TUYYHAHSIRD52LXT","created_at":"2026-05-18T12:26:02.257875+00:00"},{"alias_kind":"pith_short_8","alias_value":"TUYYHAHS","created_at":"2026-05-18T12:26:02.257875+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/TUYYHAHSIRD52LXTRKWKSTMFMS","json":"https://pith.science/pith/TUYYHAHSIRD52LXTRKWKSTMFMS.json","graph_json":"https://pith.science/api/pith-number/TUYYHAHSIRD52LXTRKWKSTMFMS/graph.json","events_json":"https://pith.science/api/pith-number/TUYYHAHSIRD52LXTRKWKSTMFMS/events.json","paper":"https://pith.science/paper/TUYYHAHS"},"agent_actions":{"view_html":"https://pith.science/pith/TUYYHAHSIRD52LXTRKWKSTMFMS","download_json":"https://pith.science/pith/TUYYHAHSIRD52LXTRKWKSTMFMS.json","view_paper":"https://pith.science/paper/TUYYHAHS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0905.4751&json=true","fetch_graph":"https://pith.science/api/pith-number/TUYYHAHSIRD52LXTRKWKSTMFMS/graph.json","fetch_events":"https://pith.science/api/pith-number/TUYYHAHSIRD52LXTRKWKSTMFMS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TUYYHAHSIRD52LXTRKWKSTMFMS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TUYYHAHSIRD52LXTRKWKSTMFMS/action/storage_attestation","attest_author":"https://pith.science/pith/TUYYHAHSIRD52LXTRKWKSTMFMS/action/author_attestation","sign_citation":"https://pith.science/pith/TUYYHAHSIRD52LXTRKWKSTMFMS/action/citation_signature","submit_replication":"https://pith.science/pith/TUYYHAHSIRD52LXTRKWKSTMFMS/action/replication_record"}},"created_at":"2026-05-18T04:11:49.344931+00:00","updated_at":"2026-05-18T04:11:49.344931+00:00"}