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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 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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 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