 Exact null distribution of a rank sum

Under the continuous common-distribution null of the Wilcoxon rank sum test, pooled labels are exchangeable. The product records every choice of ranks: choosing $z{t}^k$ assigns rank $k$ to the first sample. Its indicated coefficient counts subsets of size $m$ and rank sum $w$. Divide by $\left(\genfrac{}{}{0ex}{}{m+n}{m}\right)$ and sum the lower tail for the one-sided P-value. Equivalently the counts satisfy $a_{k,h,w}=a_{k-1,h,w}+a_{k-1,h-1,w-k}$ with $a_{0,0,0}=1$, allowing an exact integer computation.