Geodesic
A generating function for all semi-magic squares and the volume of the Birkhoff polytope
Journal of Algebraic Combinatorics, Tome 30 (2009) no. 1, pp. 113-139.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We present a multivariate generating function for all $n\times n$ nonnegative integral 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 all coefficients of the Ehrhart polynomial of the polytope $B _{ n }$ of $n\times n$ doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of $B _{ n }$ and any of its faces.
Keywords: keywords Birkhoff polytope, volume, lattice points, generating functions, Ehrhart polynomials 
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De Loera, J.A.; Liu, F.; Yoshida, R. A generating function for all semi-magic squares and the volume of the Birkhoff polytope. Journal of Algebraic Combinatorics, Tome 30 (2009) no. 1, pp. 113-139. http://geodesic.mathdoc.fr/item/JAC_2009__30_1_a0/