The moduli space of pointed rational curves has a natural action of the symmetric group permuting the marked points.  In this talk, I will present combinatorial and recursive formulas for the induced representation on the cohomology of the moduli space. These formulas are derived from wall crossings of birational models, governed by Hassett’s theory of weighted stable curves and Choi-Kiem’s theory of delta-stability of quasimaps. These results allow us to investigate positivity and log-concavity of the representation. Based on joint works with Jinwon Choi and Young-Hoon Kiem.