Jeff,

you can do

> sum1: \sum_i\sum_j max(X_i,X_j)
> sum2: \sum_i\sum_j max(Y_i,Y_j)

sum(x * (2 * rank(x) - 1))

> sum3: \sum_i\sum_j max(X_i,Y_j)

sum(outer(x, y, pmax))

Probably, the latter can be speeded up even more...
Z

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