Abstract. 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. The leading term of the Ehrhart polynomial is--up to a trivial factor--the volume of the polytope, which is one reason why we are interested in this counting function. We implemented our methods in the form of a computer program, which yielded the Ehrhart polynomial (and hence the volume) of the ninth Birkhoff polytope. Recently we also obtained the volume of the tenth Birkhoff polytope.

Here is the announcement of the calculation of the volume of the tenth Birkhoff polytope (dvi, pdf, postscript).

The accompanying C⁺⁺ program and our calculations, including the coefficients of H_n for n ≤ 9 and the volume V_n for n ≤ 10 , can be found here.