Consider a topological space $X$, a natural number $n>0$ and the quotient topological space $Q_n(X)$ of $X^n$ by the equivalence relation : $x\sim y$ if and only if there is a permutation $\sigma$ such that $x_i=y_{\sigma(i)}$.
I've seen many results on the usual unordered configuration space (where we only consider distincts points), and my question is basically : what is known about $Q_n(X)$ ? Have these spaces been studied ? For example, is $Q_n(\mathbb{R^2})$ the classifying space of a known group ? Do we know how to compute the fundamental group of $Q_n(\mathbb{R^2})$ ?
