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We prove a dimension formula for the irreducible modules of the centralizer algebra $\\mathsf{Z}_k(n)$ in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities "},"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":"1605.06543","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2016-05-20T21:35:48Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"4bec9a61a12599c28893fc28bbfd1a96cf5c68bcd78f54bedacd64f0cdcb1480","abstract_canon_sha256":"2ed4228092aa9848f12a9687892292e27dd4194d01e71bca51ed0779359e2504"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:13:10.879220Z","signature_b64":"cChIpzWGHAJDpCxNXQpR68EeDfDzYtMkCBfLwSVIQvw56mwIQKyA/LHurmwfvWylOcXxoLI+E712YknDV3FCDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3fe0d1372812386458ac0d81e7a68782ca41ea8b0a089cab25820d78dbe5145e","last_reissued_at":"2026-05-18T01:13:10.878815Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:13:10.878815Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.RT","authors_text":"Georgia Benkart, Nate Harman, Tom Halverson","submitted_at":"2016-05-20T21:35:48Z","abstract_excerpt":"The partition algebra $\\mathsf{P}_k(n)$ and the symmetric group $\\mathsf{S}_n$ are in Schur-Weyl duality on the $k$-fold tensor power $\\mathsf{M}_n^{\\otimes k}$ of the permutation module $\\mathsf{M}_n$ of $\\mathsf{S}_n$, so there is a surjection $\\mathsf{P}_k(n) \\to \\mathsf{Z}_k(n) := \\mathsf{End}_{\\mathsf{S}_n}(\\mathsf{M}_n^{\\otimes k}),$ which is an isomorphism when $n \\ge 2k$. We prove a dimension formula for the irreducible modules of the centralizer algebra $\\mathsf{Z}_k(n)$ in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.06543","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":"1605.06543","created_at":"2026-05-18T01:13:10.878874+00:00"},{"alias_kind":"arxiv_version","alias_value":"1605.06543v2","created_at":"2026-05-18T01:13:10.878874+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.06543","created_at":"2026-05-18T01:13:10.878874+00:00"},{"alias_kind":"pith_short_12","alias_value":"H7QNCNZICI4G","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_16","alias_value":"H7QNCNZICI4GIWFM","created_at":"2026-05-18T12:30:19.053100+00:00"},{"alias_kind":"pith_short_8","alias_value":"H7QNCNZI","created_at":"2026-05-18T12:30:19.053100+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/H7QNCNZICI4GIWFMBWA6PJUHQL","json":"https://pith.science/pith/H7QNCNZICI4GIWFMBWA6PJUHQL.json","graph_json":"https://pith.science/api/pith-number/H7QNCNZICI4GIWFMBWA6PJUHQL/graph.json","events_json":"https://pith.science/api/pith-number/H7QNCNZICI4GIWFMBWA6PJUHQL/events.json","paper":"https://pith.science/paper/H7QNCNZI"},"agent_actions":{"view_html":"https://pith.science/pith/H7QNCNZICI4GIWFMBWA6PJUHQL","download_json":"https://pith.science/pith/H7QNCNZICI4GIWFMBWA6PJUHQL.json","view_paper":"https://pith.science/paper/H7QNCNZI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1605.06543&json=true","fetch_graph":"https://pith.science/api/pith-number/H7QNCNZICI4GIWFMBWA6PJUHQL/graph.json","fetch_events":"https://pith.science/api/pith-number/H7QNCNZICI4GIWFMBWA6PJUHQL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/H7QNCNZICI4GIWFMBWA6PJUHQL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/H7QNCNZICI4GIWFMBWA6PJUHQL/action/storage_attestation","attest_author":"https://pith.science/pith/H7QNCNZICI4GIWFMBWA6PJUHQL/action/author_attestation","sign_citation":"https://pith.science/pith/H7QNCNZICI4GIWFMBWA6PJUHQL/action/citation_signature","submit_replication":"https://pith.science/pith/H7QNCNZICI4GIWFMBWA6PJUHQL/action/replication_record"}},"created_at":"2026-05-18T01:13:10.878874+00:00","updated_at":"2026-05-18T01:13:10.878874+00:00"}