Abstract: The moduli space of pointed rational curves carries a natural action of the symmetric group permuting the marked points. In this talk, I will present combinatorial and recursive formulas for the induced representations on its cohomology. These formulas are obtained by relating the representation to that of the moduli space with one additional marked point fixed under the symmetric group action, which has a particularly nice and tractable structure. Taking the invariant subspace, in particular, leads to an effective inductive formula for the Betti numbers of the moduli space of rational curves with unordered markings. I will also discuss several interesting conjectural properties of these representations. This talk is based on joint work with Jinwon Choi and Young-Hoon Kiem.