  We consider sequences of integers defined by a system of linear inequalities
with integer coefficients. We show that when the constraints are strong enough
to guarantee that all the entries are nonnegative, the generating function for
the integer solutions of weight $n$ has a finite product form. The results are
proved bijectively and are used to give several examples of interesting
identities for integer partitions and compositions. The method can be adapted
to accommodate equalities along with inequalities and can be used to obtain
multivariate forms of the generating function. We show how to extend the
technique to obtain the generating function when some coefficients are allowed
to be rational, generalizing the case of lecture hall partitions. Our initial
results were conjectured thanks to the Omega package.
