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In the course of discussing another MO question we realized that we did not know the answer to a more basic question, namely: Is it true that for every positive integer $k$ there exists a balanced ...

Let $f(G)$ give number of perfect matchings of a graph $G$. Denote $\mathcal N_{2n}=\{0,1,2,\dots,n!-1,n!\}$. Denote collection of all $2n$ vertex balanced bipartite graph to be $\mathcal G_{2n}$. ...