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A generating function for all semimagic squares and the volume of the Birkhoff polytope
 J. Algebraic Combin
"... Abstract. We present a multivariate generating function for all n×n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semimagic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope Bn of n×n ..."
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Abstract. We present a multivariate generating function for all n×n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semimagic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope Bn of n×n doublystochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of Bn and any of its faces. 1.
Formulas for the volumes of the polytope of doublystochastic matrices and its faces
, 2007
"... We provide an explicit combinatorial formula for the volume of the polytope of n×n doublystochastic matrices, also known as the Birkhoff polytope. We do this through the description of a generating function for all the lattice points of the closely related polytope of n×n real nonnegative matric ..."
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We provide an explicit combinatorial formula for the volume of the polytope of n×n doublystochastic matrices, also known as the Birkhoff polytope. We do this through the description of a generating function for all the lattice points of the closely related polytope of n×n real nonnegative matrices with all row and column sums equal to an integer t. We can in fact recover similar formulas for all coefficients of the Ehrhart polynomial of the Birkhoff polytope and for all its faces.