{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:EBPDHMCGS6K2PFESQRAI5I73TQ","short_pith_number":"pith:EBPDHMCG","schema_version":"1.0","canonical_sha256":"205e33b0469795a7949284408ea3fb9c074481fa0a1b5efede26ce6ab6033960","source":{"kind":"arxiv","id":"1510.04310","version":3},"attestation_state":"computed","paper":{"title":"A Fibonacci analogue of Stirling numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jeffrey B. Remmel, Quang T. Bach, Roshil Paudyal","submitted_at":"2015-10-14T20:54:16Z","abstract_excerpt":"Consider the Fibonacci numbers defined by setting $F_1=1=F_2$ and $F_n =F_{n-1}+F_{n-2}$ for $n \\geq 3$. We let $n_F! = F_1 \\cdots F_n$ and $\\binom{n}{k}_F = \\frac{n_F!}{k_F!(n-k)_F!}$. Let $(x)_{\\downarrow_0} = (x)_{\\uparrow_0} = 1$ and for $k \\geq 1$, $(x)_{\\downarrow_k} = x(x-1) \\cdots (x-k+1)$ and $(x)_{\\uparrow_k} = x(x+1) \\cdots (x+k-1)$. Then the Stirling numbers of the first and second kind are the connections coefficients between the usual power basis $\\{x^n:n \\geq 0\\}$ and the falling factorial basis $\\{(x)_{\\downarrow_n}:n \\geq 0\\}$ in the polynomial ring $\\mathbb{Q}[x]$ and the Lah"},"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":"1510.04310","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-14T20:54:16Z","cross_cats_sorted":[],"title_canon_sha256":"72fa59e05457292f937cf95bfa138e06cf4cb62cb0edd763bf041d0761c6e0a2","abstract_canon_sha256":"35bc219a1075020d73d6e317560b685ec7a922fac757b5cad4e7a8fdc2045e6b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:41.294375Z","signature_b64":"1kZIkIpoCWM0lsicfQzay5CoV/sPfJV99IGHJVkEGdWjx1bfLnt1PLhBLPJH8wLB4ARzt7oTGNeNVE5RngCZBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"205e33b0469795a7949284408ea3fb9c074481fa0a1b5efede26ce6ab6033960","last_reissued_at":"2026-05-18T01:11:41.294029Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:41.294029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Fibonacci analogue of Stirling numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jeffrey B. Remmel, Quang T. Bach, Roshil Paudyal","submitted_at":"2015-10-14T20:54:16Z","abstract_excerpt":"Consider the Fibonacci numbers defined by setting $F_1=1=F_2$ and $F_n =F_{n-1}+F_{n-2}$ for $n \\geq 3$. We let $n_F! = F_1 \\cdots F_n$ and $\\binom{n}{k}_F = \\frac{n_F!}{k_F!(n-k)_F!}$. Let $(x)_{\\downarrow_0} = (x)_{\\uparrow_0} = 1$ and for $k \\geq 1$, $(x)_{\\downarrow_k} = x(x-1) \\cdots (x-k+1)$ and $(x)_{\\uparrow_k} = x(x+1) \\cdots (x+k-1)$. Then the Stirling numbers of the first and second kind are the connections coefficients between the usual power basis $\\{x^n:n \\geq 0\\}$ and the falling factorial basis $\\{(x)_{\\downarrow_n}:n \\geq 0\\}$ in the polynomial ring $\\mathbb{Q}[x]$ and the Lah"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.04310","kind":"arxiv","version":3},"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":"1510.04310","created_at":"2026-05-18T01:11:41.294083+00:00"},{"alias_kind":"arxiv_version","alias_value":"1510.04310v3","created_at":"2026-05-18T01:11:41.294083+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.04310","created_at":"2026-05-18T01:11:41.294083+00:00"},{"alias_kind":"pith_short_12","alias_value":"EBPDHMCGS6K2","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_16","alias_value":"EBPDHMCGS6K2PFES","created_at":"2026-05-18T12:29:19.899920+00:00"},{"alias_kind":"pith_short_8","alias_value":"EBPDHMCG","created_at":"2026-05-18T12:29:19.899920+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/EBPDHMCGS6K2PFESQRAI5I73TQ","json":"https://pith.science/pith/EBPDHMCGS6K2PFESQRAI5I73TQ.json","graph_json":"https://pith.science/api/pith-number/EBPDHMCGS6K2PFESQRAI5I73TQ/graph.json","events_json":"https://pith.science/api/pith-number/EBPDHMCGS6K2PFESQRAI5I73TQ/events.json","paper":"https://pith.science/paper/EBPDHMCG"},"agent_actions":{"view_html":"https://pith.science/pith/EBPDHMCGS6K2PFESQRAI5I73TQ","download_json":"https://pith.science/pith/EBPDHMCGS6K2PFESQRAI5I73TQ.json","view_paper":"https://pith.science/paper/EBPDHMCG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1510.04310&json=true","fetch_graph":"https://pith.science/api/pith-number/EBPDHMCGS6K2PFESQRAI5I73TQ/graph.json","fetch_events":"https://pith.science/api/pith-number/EBPDHMCGS6K2PFESQRAI5I73TQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EBPDHMCGS6K2PFESQRAI5I73TQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EBPDHMCGS6K2PFESQRAI5I73TQ/action/storage_attestation","attest_author":"https://pith.science/pith/EBPDHMCGS6K2PFESQRAI5I73TQ/action/author_attestation","sign_citation":"https://pith.science/pith/EBPDHMCGS6K2PFESQRAI5I73TQ/action/citation_signature","submit_replication":"https://pith.science/pith/EBPDHMCGS6K2PFESQRAI5I73TQ/action/replication_record"}},"created_at":"2026-05-18T01:11:41.294083+00:00","updated_at":"2026-05-18T01:11:41.294083+00:00"}