The Peterson variety is a subvariety of the full flag variety, and as such possesses a cohomology class, which can be expanded in the basis of Schubert classes. The coefficients are nonnegative since they represent certain intersection numbers, but can be tricky to compute with geometric methods. Our goal is to find a combinatorial interpretation for these coefficients. In this talk, we use an algebro-combinatorial approach to give some partial answers to this problem. These turn out to reveal beautiful and intriguing combinatorics: Schubert polynomials, reduced words of permutations, and tableaux will play key roles. Joint work with Vasu Tewari (UPenn).