Symmetrization of a U-statistic kernel ($h_s(x_1,\dots ,x_r)=\frac{1}{r!}{\sum }_{\pi }h(x_{\pi (1)},\dots ,x_{\pi (r)})$)

Exchangeability of an independent identically distributed sample makes every permuted kernel have the same expectation. Averaging over permutations therefore leaves the target functional unchanged and gives a symmetric kernel. The U-statistic then averages this kernel over the unordered r-element sample subsets.