The Bruhat order on the symmetric group is a partial order which bridges combinatorics, representation theory and geometry. In this talk, we shall give an overview of the history of Bruhat orders. Then we investigate a specific family of Bruhat order intervals on the symmetric group, and prove that the Poincare polynomials of such intervals are palindromic. This settles a conjecture of Francesco Brenti, Angela Carnevale and Bridget Tenner.