For $\alpha\in\{1,\dots,n\}^n$ let ${\cal M}_\alpha$ be the class of matrices $M\in\{0,1\}^n$ which contain exactly $\alpha(i)$ times $1$ in the $i$th line.
Now I want to know which matrices $M\in{\cal M}_\alpha$ maximize the following score: $$|\{\sigma\in{\cal S}_n:\forall i.M_{i,\sigma(i)}=1\}|$$
Are there any results pointing in that direction?
[EDIT] User bof just pointed out that if we let ${\cal G}_\alpha$ be the class of (bipartite) graphs with vertex set $\{v_1,\dots,v_n,w_1,\dots,w_n\}$ and edges only between $v$s and $w$s such that $v_i$ has degree $\alpha(i)$ then we are asking for graphs $G\in{\cal G}_\alpha$ which maximize the number of perfect matchings.
Since your matrices are binary, your score equals to their permanent and an upper bound for it is given by Bregman-Minc inequality.