We study algebraic algorithms for expressing the number of non-negative
integer solutions to a unimodular system of linear equations as a function of
the right hand side. Our methods include Todd classes of toric varieties via
Gr\"obner bases, and rational generating functions as in Barvinok's algorithm.
We report polyhedral and computational results for two special cases: counting
contingency tables and Kostant's partition function.Comment: 21 page
