  The n'th Birkhoff polytope is the set of all doubly stochastic n-by-n
matrices, that is, those matrices with nonnegative real coefficients in which
every row and column sums to one. A wide open problem concerns the volumes of
these polytopes, which have been known for n up to 8. We present a new,
complex-analytic way to compute the Ehrhart polynomial of the Birkhoff
polytope, that is, the function counting the integer points in the dilated
polytope. One reason to be interested in this counting function is that the
leading term of the Ehrhart polynomial is--up to a trivial factor--the volume
of the polytope. We implemented our methods in form of a computer program,
which yielded the Ehrhart polynomial (and hence the volume) of the ninth
Birkhoff polytope.
