Journal of Algebraic Combinatorics

, Volume 30, Issue 1, pp 113–139 | Cite as

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 B n of n×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.


Birkhoff polytope Volume Lattice points Generating functions Ehrhart polynomials 


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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.University of California DavisDavisUSA

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