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

Authors:
J. A. De Loera;F. Liu;R. Yoshida
Affiliations:
University of California Davis, Davis, USA 95616;University of California Davis, Davis, USA 95616;University of California Davis, Davis, USA 95616
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2009

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Abstract

We present a multivariate generating function for all n脳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脳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.