I have a faint memory of having read about general considerations regarding the evaluation of sums of the form:
$$\sum_{i_1=1}^{U_1}\sum_{i_2=1 \\ i_2 \ne i_1}^{U_2} \cdots\sum_{i_k =1 \\ i_k \ne i_{j<k}}^{U_k} f(i_k)$$
I mean $f$ and the $U_j$'s as generally as possible, but how generally these remembered considerations could be applied, I don't know. I already have some ideas about how one would evaluate sums of the form above, but I don't want to get lost in all kinds of self-inflicted mathanigans if this path is already well-trodden.
Any pointers to references/principles for evaluations of sums of this form, or examples of specific sums like this being evaluated, would be appreciated.