In the last few years Chari's convex ear decomposition has been applied to several apparently unrelated order complexes of posets to derive new insight into their f-vectors. An unexpected consequence of these investigations is that their flag f-vectors are closely related to a largely unexplored interaction between descent sets and the weak order of finite Coxeter groups. The goal of the lecture is to explain how face enumeration on these posets naturally leads to a zoo of open problems involving descent sets and the weak order. In order to guarantee maximum accessibility I will concentrate on the special case of the symmetric group.