On permanents of random doubly stochastic matrices and on asymptotic estimates for the number of Latin rectangles and Latin squares

We consider the class $\mathfrak A_n(k)$ of all $(0,1)$-matrices $A_k$ of size $n\times n$ with exactly $k$ ones in each row and each column, $k=1,\dots,n$. We prove an asymptotic formula for the permanent $\operatorname{per}A_k$, which holds true as $n\to\infty$ and $0<n-k=o(n/\ln n)$ uniformly with respect to $A_k\in\mathfrak A_n(k)$. We discuss the known upper and lower bounds for the numbers of $m\times n$ Latin rectangles and of $n\times n$ Latin squares and asymptotic expressions of these numbers as $n\to\infty$ and $m=m(n)$. We notice that the well-known O'Neil conjecture on the asymptotic behaviour of the number of Latin squares holds in a strong form. We formulate new conjectures of such kind and deduce from these conjectures asymptotic estimates of the numbers of Latin rectangles and Latin squares that sharpen the results known before. In conclusion, we give a short review of the literature devoted to the questions discussed in the paper with formulations of the main results.