Exact null distribution of a rank sum ($\mathrm{Pr}(W_X=w)={\left(\genfrac{}{}{0ex}{}{m+n}{m}\right)}^{-1}[z^m{t}^w]{\prod }_{k=1}^{m+n}(1+z{t}^k)$)

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.

For two independent continuous samples of positive sizes $m,n$ under a common-distribution null, each $m$-subset of ranks $1,\dots ,m+n$ is equally likely. The smallest rank sum is $m(m+1)/2$, attained only by $\{1,\dots ,m\}$. The only subset with rank sum one larger is $\{1,\dots ,m-1,m+1\}$. Hence the exact lower tail at that threshold is $2/\left(\genfrac{}{}{0ex}{}{m+n}{m}\right)$. This follows directly by observing that any other change increases the sum by at least two.