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Dyadic distribution of $0/1$ permanents

Fix reals $a,b\in(1,2)$ satisfying $1<b<a<ab<2$. What fraction of $0/1$ matrices of dimensions $n\times n$ have permanents in $[b2^m,a2^m]$ at some $m\in\{0,1,2,\dots,\lfloor\log_2n!\...