The vector partition function \(p_A\) associated to a \(d \times n\) matrix A with integer entries is the function \(\mathbb {Z}^d \rightarrow \mathbb {N}\) defined by \(\mathbf{b} \rightarrow \#\{\mathbf{x} \in \mathbb {N}^n: A\mathbf{x} = \mathbf{b}\}\) . It is known that vector partition functions are piecewise quasi-polynomials whose domains of quasi-polynomiality are maximal cones (chambers) of a fan called the chamber complex of A. In this article we introduce external columns and external chambers of vector partition functions. Our main result is that (up to a saturation condition) the quasi-polynomial associated to a chamber containing k external columns arises from a vector partition function with k fewer equations and variables. In the case that the chamber is external—that is, when the number of external columns in a chamber is as large as possible without being trivial—the quasi-polynomial arises from a coin exchange problem. By exploiting this we are able to obtain a determinantal formula, characterize when the quasi-polynomial is polynomial, and show that in this case it is actually given, up to sign, by a negative binomial coefficient. We then apply these results to the enumeration of loopless multigraphs satisfying certain degree conditions. Finally, we discuss possible generalizations of our results to some problems in combinatorics and geometry.
