  The vector partition function $p_A$ associated to a $d \times n$ matrix $A$
with integer entries is the function $\mathbb{Z}^d \to \mathbb{N}$ defined by
$\mathbf{b} \to \#\{\mathbf{x} \in \mathbb{N}^n : A\mathbf{x} = \mathbf{b}\}$.
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 \emph{external columns} and \emph{external
chambers} of vector partition functions. Our main result is that (up to a
saturation condition) the quasi-polynomial associated to a chamber containing
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 by a negative binomial coefficient. We then apply
these results to the enumeration of loopless multigraphs satisfying some degree
conditions. Finally, we suggest a generalization to a result of Baldoni and
Vergne for polynomials arising from chambers that we call \emph{semi-external
chambers}.
