{"paper":{"title":"A novel permanent identity with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Wei Xia, Yue-Feng She, Zhi-Wei Sun","submitted_at":"2022-08-25T15:50:32Z","abstract_excerpt":"Let $n$ be a positive integer, and define the rational function $S(x_1,\\ldots,x_{2n})$ as the permanent of the matrix $[x_{j,k}]_{1\\le j,k\\le 2n}$, where $$x_{j,k}=\\begin{cases}(x_j+x_k)/(x_j-x_k)&\\text{if}\\ j\\not=k,\\\\1&\\text{if}\\ j=k.\\end{cases}$$ We give an explicit formula for $S(x_1,\\ldots,x_{2n})$ which has the following consequence: If one of the variables $x_1,\\ldots,x_{2n}$ takes zero, then $S(x_1,\\ldots,x_{2n})$ vanishes, i.e., $$\\sum_{\\tau\\in S_{2n}}\\prod_{j=1\\atop \\tau(j)\\not=j}^{2n}\\frac{x_j+x_{\\tau(j)}}{x_j-x_{\\tau(j)}}=0,$$ where we view an empty product $\\prod_{i\\in\\emptyset}a_i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2208.12167","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2208.12167/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"}