{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:CBY4K5CAQ2W4E7N6ZKOHJZVHTN","short_pith_number":"pith:CBY4K5CA","schema_version":"1.0","canonical_sha256":"1071c5744086adc27dbeca9c74e6a79b5af3ccd203589007ab30bdffbe46434e","source":{"kind":"arxiv","id":"1211.0652","version":1},"attestation_state":"computed","paper":{"title":"A Combinatorial Interpretation of the Joint Cumulant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Connor Ahlbach, Jeremy Usatine, Nicholas Pippenger","submitted_at":"2012-11-03T23:59:38Z","abstract_excerpt":"In this paper, we apply the combinatorial proof technique of Description, Involution, Exceptions (DIE) to prove various known identities for the joint cumulant. Consider a set of random variables $S = \\{X_1,..., X_n\\} $. Motivated by the definition of the joint cumulant, we define $ \\sC(S) $ as the set of cyclically arranged partitions of $S$, allowing us to express the joint cumulant of $ S $ as a weighted, alternating sum over $\\sC(S)$. We continue to define other combinatorial objects that allow us to rewrite expressions originally in terms of the joint cumulant as weighted sums over the se"},"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":"1211.0652","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-03T23:59:38Z","cross_cats_sorted":[],"title_canon_sha256":"625bdc55523f95831fb966bcfd63e7fdb71493449a9492840409465fa0183f80","abstract_canon_sha256":"59db45094418cef3a675a24ff331dfa77b379032e71001e16b2c2d54ac6da55f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:32.916743Z","signature_b64":"Uw78FEJeCJZpUKFcwVsJSu0G1KIx3zy8v2p4mpBN453mNF1wG4WrObPpAwxIc5kVVG0BekhR3zE/dV3S3i7gBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1071c5744086adc27dbeca9c74e6a79b5af3ccd203589007ab30bdffbe46434e","last_reissued_at":"2026-05-18T03:41:32.916016Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:32.916016Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Combinatorial Interpretation of the Joint Cumulant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Connor Ahlbach, Jeremy Usatine, Nicholas Pippenger","submitted_at":"2012-11-03T23:59:38Z","abstract_excerpt":"In this paper, we apply the combinatorial proof technique of Description, Involution, Exceptions (DIE) to prove various known identities for the joint cumulant. Consider a set of random variables $S = \\{X_1,..., X_n\\} $. Motivated by the definition of the joint cumulant, we define $ \\sC(S) $ as the set of cyclically arranged partitions of $S$, allowing us to express the joint cumulant of $ S $ as a weighted, alternating sum over $\\sC(S)$. We continue to define other combinatorial objects that allow us to rewrite expressions originally in terms of the joint cumulant as weighted sums over the se"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.0652","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":"1211.0652","created_at":"2026-05-18T03:41:32.916133+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.0652v1","created_at":"2026-05-18T03:41:32.916133+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.0652","created_at":"2026-05-18T03:41:32.916133+00:00"},{"alias_kind":"pith_short_12","alias_value":"CBY4K5CAQ2W4","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_16","alias_value":"CBY4K5CAQ2W4E7N6","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_8","alias_value":"CBY4K5CA","created_at":"2026-05-18T12:27:01.376967+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/CBY4K5CAQ2W4E7N6ZKOHJZVHTN","json":"https://pith.science/pith/CBY4K5CAQ2W4E7N6ZKOHJZVHTN.json","graph_json":"https://pith.science/api/pith-number/CBY4K5CAQ2W4E7N6ZKOHJZVHTN/graph.json","events_json":"https://pith.science/api/pith-number/CBY4K5CAQ2W4E7N6ZKOHJZVHTN/events.json","paper":"https://pith.science/paper/CBY4K5CA"},"agent_actions":{"view_html":"https://pith.science/pith/CBY4K5CAQ2W4E7N6ZKOHJZVHTN","download_json":"https://pith.science/pith/CBY4K5CAQ2W4E7N6ZKOHJZVHTN.json","view_paper":"https://pith.science/paper/CBY4K5CA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.0652&json=true","fetch_graph":"https://pith.science/api/pith-number/CBY4K5CAQ2W4E7N6ZKOHJZVHTN/graph.json","fetch_events":"https://pith.science/api/pith-number/CBY4K5CAQ2W4E7N6ZKOHJZVHTN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CBY4K5CAQ2W4E7N6ZKOHJZVHTN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CBY4K5CAQ2W4E7N6ZKOHJZVHTN/action/storage_attestation","attest_author":"https://pith.science/pith/CBY4K5CAQ2W4E7N6ZKOHJZVHTN/action/author_attestation","sign_citation":"https://pith.science/pith/CBY4K5CAQ2W4E7N6ZKOHJZVHTN/action/citation_signature","submit_replication":"https://pith.science/pith/CBY4K5CAQ2W4E7N6ZKOHJZVHTN/action/replication_record"}},"created_at":"2026-05-18T03:41:32.916133+00:00","updated_at":"2026-05-18T03:41:32.916133+00:00"}