2002/02/28 by Matthias Beck, Dennis Pixton · 1 citation
Mathematics · #math.CO #msc:05A15 #msc:52C07
published as Discrete & Computational Geometry 30, no. 4 (2003), 623-637 · 14 pages
arxiv created 2005/01/02 · arxiv updated 2009/11/30
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.