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I have tried many ways to prove the following equation, such as the method of induction and expanding all the terms in the summation,but things got more complicated.I could not find an appropriate ...

Define an antisymmetric 1-x-matrix as an $n\times n$ matrix $M=(m_{ij})$ with $m_{ii}=0$ and $\{m_{ij},m_{ji}\}=\{1,x\}$ for all $1\le i<j\le n$. Call their set $\mathcal A_n$. The setup is as ...