ABSTRACT: How many faces in each dimension can a simplicial polytope have? This question turns out to have a beautiful answer, which was conjectured by McMullen in 1971 and proved by Stanley and Billera-Lee in the 80s. It is natural to ask what one can say for other combinatorial structures. I will review several instances where hard Lefschetz provides the only known explanation of “top heaviness”. Then I will focus on Bruhat intervals, where only pieces of the puzzle are understood.