Combinatorics, computer algebra and Wilcoxon-Mann-Whitney test

We show the combinatorics behind the Wilcoxon-??Mann-??Whitney two-??sample test.?? This yields new combinatorial proofs of recurrences for its null distribution given recently by Brus and Chang??, as well as new recurrences.?? It is shown how to convert these recurrences into generating functions.?? These generating functions are used to obtain closed expressions for the null distribution when one of the sample sizes is fixed and to compute moments??. We also show how to perform these calculations with the aid of the computer algebra system Mathematica??. Keywords: Wilcoxon??-Mann-??Whitney two-??sample test,?? partitions,?? Gaussian binomial coefficients,?? computer algebra,?? generating functions,?? recurrences.