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We present a multivariate generating function for all n x n nonnegativeintegral matrices with all row and column sums equal to a positive integer t,the so called semi-magic squares. As a consequence we obtain formulas for allcoefficients of the Ehrhart polynomial of the polytope B_n of n x ndoubly-stochastic matrices, also known as the Birkhoff polytope. In particularwe derive formulas for the volumes of B_n and any of its faces.Comment: 24 pages, 1 figure. To appear in Journal of Algebraic Combinatoric

A generating function for all semi-magic squares and the volume of the Birkhoff polytope