Consider $2n$ vertex balanced bipartite graph.

If total number of edges in it is $n^2$ then we have $n!$ perfect matchings.

Suppose if $c\in(0,\frac12)$ is fixed and total number of edges in it is $cn^2$ then should it have at least $f(c)n!$ perfect matchings for some function $f:\Bbb R\rightarrow\Bbb R$?