The vector partition problem with restricted parts, first considered by Blakely in 1964, is a generalization of the classic problem in combinatorics of enumerating the number of partitions of some integer with restricted parts. Here we are given a multi-set S of n vectors in Z^m and asked to find the number of ways that an arbitrary vector b in Z^m can be written as a linear combination of the elements of S with non-negative integer coefficients. It was proven by Sturmfels that the number of such solutions is a piecewise quasi-polynomial p_S. Furthermore, the domains of quasi-polynomiality are elements of the chamber complex of the m x n matrix A whose columns are the vectors of S.
We apply techniques given in the seminal text “Analytic Combinatorics in Several Variables” by Pemantle and Wilson in order to compute asymptotics for p_S. We present applications of these techniques to the study of Kronecker coefficients, and enumeration of Magic squares. We also hint at future work in the study of Littlewood-Richardson coefficients.