{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:74JKA6CG5R5IMOWTIWZYBS3OVB","short_pith_number":"pith:74JKA6CG","schema_version":"1.0","canonical_sha256":"ff12a07846ec7a863ad345b380cb6ea87790c24d13a031aa0833e997ef261183","source":{"kind":"arxiv","id":"1105.6270","version":2},"attestation_state":"computed","paper":{"title":"Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AG","math.MP"],"primary_cat":"math.CO","authors_text":"Alan D. Sokal, Andrea Sportiello, Sergio Caracciolo","submitted_at":"2011-05-31T13:25:10Z","abstract_excerpt":"The classic Cayley identity states that \\det(\\partial) (\\det X)^s = s(s+1)...(s+n-1) (\\det X)^{s-1} where X=(x_{ij}) is an n-by-n matrix of indeterminates and \\partial=(\\partial/\\partial x_{ij}) is the corresponding matrix of partial derivatives. In this paper we present straightforward combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterior algebra) and Grassmann-Berezin integration. Among the new identities proven here are a pair of \"diagonal-parametrized\" Cayley identities, a pair of \"Laplacian-par"},"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":"1105.6270","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-05-31T13:25:10Z","cross_cats_sorted":["math-ph","math.AG","math.MP"],"title_canon_sha256":"c88a7cc6f6c00f24a16ef436b3c625e90f611a2af44318aaa2ff8a85c6b9784c","abstract_canon_sha256":"89bcafc9e98f57ae783cb88a7b7668f829ee9f9e08026ce125c0cc4c76b26b40"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:17:34.443931Z","signature_b64":"j8Nnv0obkIqxbPqRBEX20znMVNerz1fm4gIwwV1UDU2ConfkxC+zMqw6luC0nBwwa0z/Hn7velnNl9qHXctEAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ff12a07846ec7a863ad345b380cb6ea87790c24d13a031aa0833e997ef261183","last_reissued_at":"2026-05-18T03:17:34.443419Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:17:34.443419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AG","math.MP"],"primary_cat":"math.CO","authors_text":"Alan D. Sokal, Andrea Sportiello, Sergio Caracciolo","submitted_at":"2011-05-31T13:25:10Z","abstract_excerpt":"The classic Cayley identity states that \\det(\\partial) (\\det X)^s = s(s+1)...(s+n-1) (\\det X)^{s-1} where X=(x_{ij}) is an n-by-n matrix of indeterminates and \\partial=(\\partial/\\partial x_{ij}) is the corresponding matrix of partial derivatives. In this paper we present straightforward combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterior algebra) and Grassmann-Berezin integration. Among the new identities proven here are a pair of \"diagonal-parametrized\" Cayley identities, a pair of \"Laplacian-par"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1105.6270","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":"1105.6270","created_at":"2026-05-18T03:17:34.443504+00:00"},{"alias_kind":"arxiv_version","alias_value":"1105.6270v2","created_at":"2026-05-18T03:17:34.443504+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1105.6270","created_at":"2026-05-18T03:17:34.443504+00:00"},{"alias_kind":"pith_short_12","alias_value":"74JKA6CG5R5I","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_16","alias_value":"74JKA6CG5R5IMOWT","created_at":"2026-05-18T12:26:22.705136+00:00"},{"alias_kind":"pith_short_8","alias_value":"74JKA6CG","created_at":"2026-05-18T12:26:22.705136+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/74JKA6CG5R5IMOWTIWZYBS3OVB","json":"https://pith.science/pith/74JKA6CG5R5IMOWTIWZYBS3OVB.json","graph_json":"https://pith.science/api/pith-number/74JKA6CG5R5IMOWTIWZYBS3OVB/graph.json","events_json":"https://pith.science/api/pith-number/74JKA6CG5R5IMOWTIWZYBS3OVB/events.json","paper":"https://pith.science/paper/74JKA6CG"},"agent_actions":{"view_html":"https://pith.science/pith/74JKA6CG5R5IMOWTIWZYBS3OVB","download_json":"https://pith.science/pith/74JKA6CG5R5IMOWTIWZYBS3OVB.json","view_paper":"https://pith.science/paper/74JKA6CG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1105.6270&json=true","fetch_graph":"https://pith.science/api/pith-number/74JKA6CG5R5IMOWTIWZYBS3OVB/graph.json","fetch_events":"https://pith.science/api/pith-number/74JKA6CG5R5IMOWTIWZYBS3OVB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/74JKA6CG5R5IMOWTIWZYBS3OVB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/74JKA6CG5R5IMOWTIWZYBS3OVB/action/storage_attestation","attest_author":"https://pith.science/pith/74JKA6CG5R5IMOWTIWZYBS3OVB/action/author_attestation","sign_citation":"https://pith.science/pith/74JKA6CG5R5IMOWTIWZYBS3OVB/action/citation_signature","submit_replication":"https://pith.science/pith/74JKA6CG5R5IMOWTIWZYBS3OVB/action/replication_record"}},"created_at":"2026-05-18T03:17:34.443504+00:00","updated_at":"2026-05-18T03:17:34.443504+00:00"}