{"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"}