A combinatorial approach to the enumeration of doubly stochastic square matrices with nonnegative integer elements

Let $H_R(n,r)$ be equal to the number of $n\times n$ matrices with non-negative integer elements such that all row sums and all column sums are equal to $r$ and all elements with indices from a set $R$ are equal to zero. We investigate the properties of the function $H_R(n,r)$ and give a combinatorial interpretation of the obtained results.