I will talk about a new basis of the polynomial ring that refines Schubert polynomials. This basis is inspired by work of Aval-Bergeron-Bergeron in the context of reducing polynomials modulo the ideal of positive degree quasisymmetric polyomials. We use this to find a manifestly nonnegative integral expansion for the class of the permutahedral variety in terms of Schubert classes, with a novel parking procedure playing a key role.