Collapsibility of $\Delta(\Pi _n)/\mathcal{S}_n$ and some related CW complexes

Let $\Delta(\Pi _n)$ denote the order complex of the partition lattice. The natural $\mathcal{S}_n$-action on the set $[n]$ induces an $\mathcal{S}_n$-action on $\Delta(\Pi _n)$. We show that the regular CW complex $\Delta (\Pi _n)/\mathcal{S}_n$ is collapsible. Even more, we show that $\Delta (\Pi _n)/\mathcal{S}_n$ is collapsible, where $\Pi _\Delta$ is a suitable type selection of the partition lattice. This allows us to generalize and reprove in a conceptual way several previous results regarding the multiplicity of the trivial character in the $\mathcal{S}_n$-representation on $H_*(\Delta(\Pi _n))$.