Gantmacher's study of totally nonnegative (TNN) matrices in the 1930's eventually found applications in many areas of mathematics. Descending from his work are problems concerning TNN polynomials, those polynomial functions of n^2 variables which take nonnegative values on TNN matrices. Closely related to TNN polynomials are functions in the Hecke algebra trace space whose evaluations at certain Hecke algebra elements yield polynomials in N[q]. In all cases, it would be desirable to combinatorially interpret the resulting nonnegative numbers. In 2017, Kaliszewski, Lambright, and the presenter found the first cancellation-free combinatorial formula for the evaluation of all elements of a basis of V at all elements of a basis of the Hecke algebra. We will discuss a recent improvement upon this result which also advances our understanding of TNN polynomials. This is joint work with Adam Clearwater.