The problem of extracting the coefficients of a generating function F(z_1 , ... , z_d) is essential to the successful analysis of many combinatorial problems. Analytic machinery exists to aid in this task. In dimension 1 (d=1), this leads to a well understood, automatic means of obtaining asymptotic formulae. In dimensions great than 1, the analogues of these results may be obtained via Stratified Morse Theory. This, however, leaves a host of open questions of effective combinatorial computation with respect to homology bases, whose existence is dictated by the theory but whose relation to specific algebraic-geometric cycles of interest is not, a priori, known.