Journal of Algebraic Combinatorics

, Volume 30, Issue 1, pp 113–139

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



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 semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope Bn of n×n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of Bn and any of its faces.


Birkhoff polytope Volume Lattice points Generating functions Ehrhart polynomials 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.University of California DavisDavisUSA

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