The (type A) Hecke algebra H_n(q) is an n!-dimensional q-analog of the symmetric group. A related trace space of certain functions on H_n(q) has dimension equal to the number of integer partitions of n. If we could evaluate all functions belonging to some basis of the trace space on all elements of some basis of H_n(q), then by linearity we could evaluate em all traces on all elements of H_n(q). Unfortunately there is no simple published formula which accomplishes this. We will consider a basis of H_n(q) which is related to structures called wiring diagrams, and a combinatorial rule for evaluating one trace basis on all elements of this wiring diagram basis. This result, the first of its kind, is joint work with Justin Lambright and Ryan Kaliszewski.