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Taking $P$ to be the intersection poset of a hyperplane arrangement $\\mathcal{A}$, we shall obtain a convolution identity for the number $r(\\mathcal{A})$ of regions and the number $b(\\mathcal{A})$ of relatively bounded regions, and a reciprocity theorem of the characteristic polynomial $\\chi(\\mathcal{A},t"},"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":"1209.2769","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-13T03:11:02Z","cross_cats_sorted":[],"title_canon_sha256":"972aad0b33c2e2cc0b21bc87572a73aa9eaee353953d40959f6b14992e2130f1","abstract_canon_sha256":"ca7248980bf20f473cb498f99f5a2c90436f6a55798f76a9e63e088b70ea0470"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:04.350572Z","signature_b64":"RpxTnRoOkApbr8w60wUhbcZx2WKhsJDGGXt5n+AsJ6SlYH0RJKwCBiTUmRkZTqoUA6/P3t3Tg1cvlqB5asz9BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9bcbcdf0b4576755a9a44778ce8418af264fc1a99c2f79370349a43317ecd81f","last_reissued_at":"2026-05-18T00:36:04.349937Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:04.349937Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Mobius Conjugation and Convolution Formulae","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Suijie Wang","submitted_at":"2012-09-13T03:11:02Z","abstract_excerpt":"Let $P$ be a locally finite poset with the interval space $\\Int(P)$, and $R$ a ring with identity. 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Taking $P$ to be the intersection poset of a hyperplane arrangement $\\mathcal{A}$, we shall obtain a convolution identity for the number $r(\\mathcal{A})$ of regions and the number $b(\\mathcal{A})$ of relatively bounded regions, and a reciprocity theorem of the characteristic polynomial $\\chi(\\mathcal{A},t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.2769","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":"1209.2769","created_at":"2026-05-18T00:36:04.350045+00:00"},{"alias_kind":"arxiv_version","alias_value":"1209.2769v1","created_at":"2026-05-18T00:36:04.350045+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.2769","created_at":"2026-05-18T00:36:04.350045+00:00"},{"alias_kind":"pith_short_12","alias_value":"TPF434FUK5TV","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_16","alias_value":"TPF434FUK5TVLKNE","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_8","alias_value":"TPF434FU","created_at":"2026-05-18T12:27:23.164592+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/TPF434FUK5TVLKNEI54M5BAYV4","json":"https://pith.science/pith/TPF434FUK5TVLKNEI54M5BAYV4.json","graph_json":"https://pith.science/api/pith-number/TPF434FUK5TVLKNEI54M5BAYV4/graph.json","events_json":"https://pith.science/api/pith-number/TPF434FUK5TVLKNEI54M5BAYV4/events.json","paper":"https://pith.science/paper/TPF434FU"},"agent_actions":{"view_html":"https://pith.science/pith/TPF434FUK5TVLKNEI54M5BAYV4","download_json":"https://pith.science/pith/TPF434FUK5TVLKNEI54M5BAYV4.json","view_paper":"https://pith.science/paper/TPF434FU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1209.2769&json=true","fetch_graph":"https://pith.science/api/pith-number/TPF434FUK5TVLKNEI54M5BAYV4/graph.json","fetch_events":"https://pith.science/api/pith-number/TPF434FUK5TVLKNEI54M5BAYV4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TPF434FUK5TVLKNEI54M5BAYV4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TPF434FUK5TVLKNEI54M5BAYV4/action/storage_attestation","attest_author":"https://pith.science/pith/TPF434FUK5TVLKNEI54M5BAYV4/action/author_attestation","sign_citation":"https://pith.science/pith/TPF434FUK5TVLKNEI54M5BAYV4/action/citation_signature","submit_replication":"https://pith.science/pith/TPF434FUK5TVLKNEI54M5BAYV4/action/replication_record"}},"created_at":"2026-05-18T00:36:04.350045+00:00","updated_at":"2026-05-18T00:36:04.350045+00:00"}