{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:THUS4U57L7QLPRS5BF3YOT7WE3","short_pith_number":"pith:THUS4U57","schema_version":"1.0","canonical_sha256":"99e92e53bf5fe0b7c65d0977874ff626df173a2fc21f035882ae454b2e1e214b","source":{"kind":"arxiv","id":"1506.03249","version":2},"attestation_state":"computed","paper":{"title":"q-Stirling numbers: A new view","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Margaret A. Readdy, Yue Cai","submitted_at":"2015-06-10T10:37:26Z","abstract_excerpt":"We show the classical $q$-Stirling numbers of the second kind can be expressed compactly as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in $q$ and $1+q$. We extend this enumerative result via a decomposition of a new poset $\\Pi(n,k)$ which we call the Stirling poset of the second kind. Its rank generating function is the $q$-Stirling number $S_q[n,k]$. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the "},"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":"1506.03249","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-06-10T10:37:26Z","cross_cats_sorted":[],"title_canon_sha256":"663ba20bd485913fc44f48ed129c3620a8b4f0945aedba360f569398016a914c","abstract_canon_sha256":"a17d6d129ed03ebb9bc70248d2a5da8095c0901acd78e6b75560ae9092e9aef9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:43:36.107753Z","signature_b64":"dNDrPH0GZR5SWAq4Vq06+xoYSid3HQSUveExzdxw+ilOJqL2ZqGu8zLZLENAEGtRDLdXuabnti7BMHS/l1d6AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99e92e53bf5fe0b7c65d0977874ff626df173a2fc21f035882ae454b2e1e214b","last_reissued_at":"2026-05-18T00:43:36.107350Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:43:36.107350Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"q-Stirling numbers: A new view","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Margaret A. Readdy, Yue Cai","submitted_at":"2015-06-10T10:37:26Z","abstract_excerpt":"We show the classical $q$-Stirling numbers of the second kind can be expressed compactly as a pair of statistics on a subset of restricted growth words. The resulting expressions are polynomials in $q$ and $1+q$. We extend this enumerative result via a decomposition of a new poset $\\Pi(n,k)$ which we call the Stirling poset of the second kind. Its rank generating function is the $q$-Stirling number $S_q[n,k]$. The Stirling poset of the second kind supports an algebraic complex and a basis for integer homology is determined. A parallel enumerative, poset theoretic and homological study for the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.03249","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":"1506.03249","created_at":"2026-05-18T00:43:36.107418+00:00"},{"alias_kind":"arxiv_version","alias_value":"1506.03249v2","created_at":"2026-05-18T00:43:36.107418+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.03249","created_at":"2026-05-18T00:43:36.107418+00:00"},{"alias_kind":"pith_short_12","alias_value":"THUS4U57L7QL","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_16","alias_value":"THUS4U57L7QLPRS5","created_at":"2026-05-18T12:29:42.218222+00:00"},{"alias_kind":"pith_short_8","alias_value":"THUS4U57","created_at":"2026-05-18T12:29:42.218222+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/THUS4U57L7QLPRS5BF3YOT7WE3","json":"https://pith.science/pith/THUS4U57L7QLPRS5BF3YOT7WE3.json","graph_json":"https://pith.science/api/pith-number/THUS4U57L7QLPRS5BF3YOT7WE3/graph.json","events_json":"https://pith.science/api/pith-number/THUS4U57L7QLPRS5BF3YOT7WE3/events.json","paper":"https://pith.science/paper/THUS4U57"},"agent_actions":{"view_html":"https://pith.science/pith/THUS4U57L7QLPRS5BF3YOT7WE3","download_json":"https://pith.science/pith/THUS4U57L7QLPRS5BF3YOT7WE3.json","view_paper":"https://pith.science/paper/THUS4U57","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1506.03249&json=true","fetch_graph":"https://pith.science/api/pith-number/THUS4U57L7QLPRS5BF3YOT7WE3/graph.json","fetch_events":"https://pith.science/api/pith-number/THUS4U57L7QLPRS5BF3YOT7WE3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/THUS4U57L7QLPRS5BF3YOT7WE3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/THUS4U57L7QLPRS5BF3YOT7WE3/action/storage_attestation","attest_author":"https://pith.science/pith/THUS4U57L7QLPRS5BF3YOT7WE3/action/author_attestation","sign_citation":"https://pith.science/pith/THUS4U57L7QLPRS5BF3YOT7WE3/action/citation_signature","submit_replication":"https://pith.science/pith/THUS4U57L7QLPRS5BF3YOT7WE3/action/replication_record"}},"created_at":"2026-05-18T00:43:36.107418+00:00","updated_at":"2026-05-18T00:43:36.107418+00:00"}