This paper gives an exposition of well known results on vector partition
functions. The exposition is based on works of M. Brion, A. Szenes and M.
Vergne and is geared toward explicit computer realizations. In particular, the
paper presents two algorithms for computing the vector partition function with
respect to a finite set of vectors $I$ as a quasipolynomial over a finite set
of pointed polyhedral cones. We use the developed techniques to relate a result
of P. Tumarkin and A. Felikson (and present an independent proof in the
particular case of finite-dimensional root systems) to give bounds for the
periods of the Kostant partition functions of $E_6$, $E_7$, $E_8$, $F_4$, $G_2$
(the periods are divisors of respectively 6, 12, 60, 12, 6).

The first of the described algorithms has been realized and is publicly
available under the Library General Public License v3.0 at
{this http URL}. We include (non-unique) partial
fraction decompositions for the generating functions of the Kostant partition
function for $A_2$, $A_3$, $A_4$, $B_2$, $B_3$, $C_2$, $C_3$, $G_2$ in the
appendix.