We provide new bounds between a combinatorial statistic of the form \(\varvec{Y =\sum _{i=1}^n X_{i,\pi (i)}}\) and the standard normal distribution, where \(\varvec{\{X_{i,j} \}_{i,j=1}^n}\) are independent real valued random variables, and \(\varvec{\pi \in \mathcal {S}_n}\) , which is independent of \(\varvec{X_{i,j}}\) and follows either the uniform or Ewens distribution. The family of the Ewens distributions appears in the context of population genetics in biology. The bounds are based on both \(\varvec{L^1}\) and \(\varvec{L}^{\varvec{\infty }}\) distances under different assumptions. As our method, we apply the approximate zero bias approach via Stein’s method to obtain the bounds.
