  We present new explicit upper bounds for the smoothness of the distribution
of the random diagonal sum $S_n=\sum_{j=1}^nX_{j,\pi(j)}$ of a random $n\times
n$ matrix $X=(X_{j,r})$, where the $X_{j,r}$ are independent integer valued
random variables, and $\pi$ denotes a uniformly distributed random permutation
on $\{1,\dots,n\}$ independent of $X$. As a measure of smoothness, we consider
the total variation distance between the distributions of $S_n$ and $1+S_n$.
Our approach uses a new auxiliary inequality for a generalized normalized
matrix hafnian, which could be of independent interest. This approach is also
used to prove upper bounds of the L\'{e}vy concentration function of $S_n$ in
the case of independent real valued random variables $X_{j,r}$.
