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Let $F_i$ denote the subset of $\mathbb{R}^n$ consisting of tuples $(x_1,\dots,x_n)$ where there are less than or equal to $i$ unique entries. Is there a computation of the homology of $\mathbb{R}^n - F_i$ in the literature? The case $i=n-1$ is done by Cohen. Perhaps the same approach would suffice: to study the map $\mathbb{R}^n - F_i \rightarrow \mathbb{R}^{n-1} - F_{i-1}$, but this is not even a fibration as it is in Cohen's case.

I have seen Li and Volic have a paper "Partial configuration spaces as pullbacks of diagrams of configuration spaces" about these, where it appears they calculate some of this homology. Is it right to say that the general answer is unknown?

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