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How to prove polynomial inequality encoded from 1-factors in $K_{2n}$

Let $G=(V,E)$ be a complete graph $K_{2n}$ and it has $m$ 1-factors $f_{i,(i=1,\dots,m)}$, where $m=\frac{(2n)!}{n!2^n}$. Some definition: $F=\{f_{1},f_{2},...,f_{2n-1}\}$ is one 1-factorization in $...
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