A generating function for all semi-magic squares and the volume of the Birkhoff polytope
- First Online:
- Cite this article as:
- De Loera, J.A., Liu, F. & Yoshida, R. J Algebr Comb (2009) 30: 113. doi:10.1007/s10801-008-0155-y
- 87 Downloads
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.