The Ehrhart polynomial of the Birkhoff polytope

Details The Ehrhart polynomial of the Birkhoff polytope The Ehrhart polynomial of the Birkhoff polytope

2002-02-25

arxiv.org/abs/math/0202267v2

Discrete & Computational Geometry 30, no. 4 (2003), 623-637

Mathematics

Combinatorics

14 pages

Scientific paper

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.

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