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We prove a $q$-analogue of Wilson's congruence $$ {\\mathfrak f}_{n-1}(q)\\equiv\\mu(n)\\pmod{\\Phi_n(q)}, $$ where $\\mu$ denotes the M\\\"obius function and $\\Phi_n(q)$ is the $n$-th cyclotomic polynomial."},"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":"1904.08857","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-18T15:57:15Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"7ee63780e2f41a6de2c70bc0f3d346de0a3bfc0c280236acfe5b2821a0164b3e","abstract_canon_sha256":"a603773b1532b05fae162a0e95ce8721a5f507d7ddc3f469556b323ab4117c7d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:12.738831Z","signature_b64":"ZMEG1JykIGykKvLpvChgqZK0CfjOzI14eorRih/4l0PZ5hM2n9bu/cPXXoHu5niy0oJBYJxt73HpFSsWxwYOCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ac01f774728c686eaf60ebbc46b4a06fd10e9d9a52d772f5c9b406c8c7b3fed8","last_reissued_at":"2026-05-17T23:48:12.738215Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:12.738215Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A $q$-analogue of Wilson's congruence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Hao Pan, Yu-Chen Sun","submitted_at":"2019-04-18T15:57:15Z","abstract_excerpt":"Let ${\\mathcal C}_n$ be the set of all permutation cycles of length $n$ over $\\{1,2,\\ldots,n\\}$. Let $${\\mathfrak f}_n(q):=\\sum_{\\sigma\\in{\\mathcal C}_{n+1}}q^{{\\mathrm maj}\\,\\sigma} $$ be a $q$-analogue of the factorial $n!$, where ${\\mathrm maj}$ denotes the major index. We prove a $q$-analogue of Wilson's congruence $$ {\\mathfrak f}_{n-1}(q)\\equiv\\mu(n)\\pmod{\\Phi_n(q)}, $$ where $\\mu$ denotes the M\\\"obius function and $\\Phi_n(q)$ is the $n$-th cyclotomic polynomial."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.08857","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":"1904.08857","created_at":"2026-05-17T23:48:12.738295+00:00"},{"alias_kind":"arxiv_version","alias_value":"1904.08857v1","created_at":"2026-05-17T23:48:12.738295+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.08857","created_at":"2026-05-17T23:48:12.738295+00:00"},{"alias_kind":"pith_short_12","alias_value":"VQA7O5DSRRUG","created_at":"2026-05-18T12:33:30.264802+00:00"},{"alias_kind":"pith_short_16","alias_value":"VQA7O5DSRRUG5L3A","created_at":"2026-05-18T12:33:30.264802+00:00"},{"alias_kind":"pith_short_8","alias_value":"VQA7O5DS","created_at":"2026-05-18T12:33:30.264802+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/VQA7O5DSRRUG5L3A5O6ENNFAN7","json":"https://pith.science/pith/VQA7O5DSRRUG5L3A5O6ENNFAN7.json","graph_json":"https://pith.science/api/pith-number/VQA7O5DSRRUG5L3A5O6ENNFAN7/graph.json","events_json":"https://pith.science/api/pith-number/VQA7O5DSRRUG5L3A5O6ENNFAN7/events.json","paper":"https://pith.science/paper/VQA7O5DS"},"agent_actions":{"view_html":"https://pith.science/pith/VQA7O5DSRRUG5L3A5O6ENNFAN7","download_json":"https://pith.science/pith/VQA7O5DSRRUG5L3A5O6ENNFAN7.json","view_paper":"https://pith.science/paper/VQA7O5DS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1904.08857&json=true","fetch_graph":"https://pith.science/api/pith-number/VQA7O5DSRRUG5L3A5O6ENNFAN7/graph.json","fetch_events":"https://pith.science/api/pith-number/VQA7O5DSRRUG5L3A5O6ENNFAN7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VQA7O5DSRRUG5L3A5O6ENNFAN7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VQA7O5DSRRUG5L3A5O6ENNFAN7/action/storage_attestation","attest_author":"https://pith.science/pith/VQA7O5DSRRUG5L3A5O6ENNFAN7/action/author_attestation","sign_citation":"https://pith.science/pith/VQA7O5DSRRUG5L3A5O6ENNFAN7/action/citation_signature","submit_replication":"https://pith.science/pith/VQA7O5DSRRUG5L3A5O6ENNFAN7/action/replication_record"}},"created_at":"2026-05-17T23:48:12.738295+00:00","updated_at":"2026-05-17T23:48:12.738295+00:00"}