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Via the Engenvector-eigenvalue Identity, we show that $$\\sum_{\\tau\\in D(n-1)}\\mathrm{sign}(\\tau)\\prod_{j=1}^{n-1}\\frac{1+\\zeta^{j-\\tau(j)}}{1-\\zeta^{j-\\tau(j)}} =(-1)^{\\frac{n-1}{2}}\\frac{((n-2)!!)^2}{n}, $$ where $D(n-1)$ is the set of all derangements of $1,\\ldots,n-1$. This confirms a previous conjecture of Z.-W. Sun. Moreover, for each $\\delta=0,1$ we determine the value of $\\det[x+m_{jk}]_{1\\le j,k\\le n}$ completely, where $$m_{jk}=\\begin{cases}(1+\\zeta^{j-k})/(1-\\zeta^{j-k})&\\text{if}\\ j\\not"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2206.02589","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-06-06T14:32:58Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"19fc2731e2d14c7db0fd90b7fe1fa8802d0f4a6547561f4138b47b66d8a28d63","abstract_canon_sha256":"e548177c1365d3961400c0d683bd3ebd0cf3a49797c14f5ea147f5c2503d17e3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:36:23.266722Z","signature_b64":"eMDdNxO3epgWKtx1D3jzOd34gJjjJ9JOQnQB3QB9JUWkSsVM3ISynBS9Ci5b/bttVoVtRUDZoUclTBEMUqp6Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0f18236071d3e09d54fbcfa73040bb966b3667bdce4b4b8734fbcc02768dcff","last_reissued_at":"2026-07-05T04:36:23.266303Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:36:23.266303Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of a conjecture involving derangements and roots of unity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Han Wang, Zhi-Wei Sun","submitted_at":"2022-06-06T14:32:58Z","abstract_excerpt":"Let $n>1$ be an odd integer. For any primitive $n$-th root $\\zeta$ of unity in the complex field. Via the Engenvector-eigenvalue Identity, we show that $$\\sum_{\\tau\\in D(n-1)}\\mathrm{sign}(\\tau)\\prod_{j=1}^{n-1}\\frac{1+\\zeta^{j-\\tau(j)}}{1-\\zeta^{j-\\tau(j)}} =(-1)^{\\frac{n-1}{2}}\\frac{((n-2)!!)^2}{n}, $$ where $D(n-1)$ is the set of all derangements of $1,\\ldots,n-1$. This confirms a previous conjecture of Z.-W. Sun. Moreover, for each $\\delta=0,1$ we determine the value of $\\det[x+m_{jk}]_{1\\le j,k\\le n}$ completely, where $$m_{jk}=\\begin{cases}(1+\\zeta^{j-k})/(1-\\zeta^{j-k})&\\text{if}\\ j\\not"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2206.02589","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2206.02589/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2206.02589","created_at":"2026-07-05T04:36:23.266363+00:00"},{"alias_kind":"arxiv_version","alias_value":"2206.02589v3","created_at":"2026-07-05T04:36:23.266363+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2206.02589","created_at":"2026-07-05T04:36:23.266363+00:00"},{"alias_kind":"pith_short_12","alias_value":"6DYYENQHDU7A","created_at":"2026-07-05T04:36:23.266363+00:00"},{"alias_kind":"pith_short_16","alias_value":"6DYYENQHDU7ATVKP","created_at":"2026-07-05T04:36:23.266363+00:00"},{"alias_kind":"pith_short_8","alias_value":"6DYYENQH","created_at":"2026-07-05T04:36:23.266363+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/6DYYENQHDU7ATVKPXT5HGBALXF","json":"https://pith.science/pith/6DYYENQHDU7ATVKPXT5HGBALXF.json","graph_json":"https://pith.science/api/pith-number/6DYYENQHDU7ATVKPXT5HGBALXF/graph.json","events_json":"https://pith.science/api/pith-number/6DYYENQHDU7ATVKPXT5HGBALXF/events.json","paper":"https://pith.science/paper/6DYYENQH"},"agent_actions":{"view_html":"https://pith.science/pith/6DYYENQHDU7ATVKPXT5HGBALXF","download_json":"https://pith.science/pith/6DYYENQHDU7ATVKPXT5HGBALXF.json","view_paper":"https://pith.science/paper/6DYYENQH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2206.02589&json=true","fetch_graph":"https://pith.science/api/pith-number/6DYYENQHDU7ATVKPXT5HGBALXF/graph.json","fetch_events":"https://pith.science/api/pith-number/6DYYENQHDU7ATVKPXT5HGBALXF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6DYYENQHDU7ATVKPXT5HGBALXF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6DYYENQHDU7ATVKPXT5HGBALXF/action/storage_attestation","attest_author":"https://pith.science/pith/6DYYENQHDU7ATVKPXT5HGBALXF/action/author_attestation","sign_citation":"https://pith.science/pith/6DYYENQHDU7ATVKPXT5HGBALXF/action/citation_signature","submit_replication":"https://pith.science/pith/6DYYENQHDU7ATVKPXT5HGBALXF/action/replication_record"}},"created_at":"2026-07-05T04:36:23.266363+00:00","updated_at":"2026-07-05T04:36:23.266363+00:00"}