On the error bound in a combinatorial central limit theorem

Let $\mathbb{X}=\{X_{ij}: 1\le i,j\le n\}$ be an $n\times n$ array of independent random variables where $n\ge2$. Let $\pi$ be a uniform random permutation of $\{1,2,\dots,n\}$, independent of $\mathbb{X}$, and let $W=\sum_{i=1}^nX_{i\pi(i)}$. Suppose $\mathbb{X}$ is standardized so that ${\mathbb{E}}W=0,\operatorname {Var}(W)=1$. We prove that the Kolmogorov distance between the distribution of $W$ and the standard normal distribution is bounded by $451\sum_{i,j=1}^n{\mathbb{E}}|X_{ij}|^3/n$. Our approach is by Stein's method of exchangeable pairs and the use of a concentration inequality.