A Generating Function for all Semi-Magic Squares and the Volume of the
  Birkhoff Polytope

A. Schrijver; A.I. Barvinok; A.I. Barvinok; A.I. Barvinok; B. Sturmfels; C.S. Chan; D. Zeilberger; D.P. Chan; E. Ehrhart; F. Liu; G. Kuperberg; G.M. Ziegler; J. A. De Loera; J. Lawrence; J.A. Loera De; M. Beck; M. Brion; P. Diaconis; P. Filliman; R. Yoshida; R.P. Stanley; V. Baldoni; V.A. Yemelichev; W. Fulton

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A Generating Function for all Semi-Magic Squares and the Volume of the Birkhoff Polytope

Authors: A. Schrijver
A.I. Barvinok
A.I. Barvinok
A.I. Barvinok
B. Sturmfels
C.S. Chan
D. Zeilberger
D.P. Chan
E. Ehrhart
F. Liu
G. Kuperberg
G.M. Ziegler
J. A. De Loera
J. Lawrence
J.A. Loera De
M. Beck
M. Brion
P. Diaconis
P. Filliman
R. Yoshida
R.P. Stanley
V. Baldoni
V.A. Yemelichev
W. Fulton
Publication date: 29 January 2007
Publisher: 'Springer Science and Business Media LLC'
Doi

Abstract

We present a multivariate generating function for all n x 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 x 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.Comment: 24 pages, 1 figure. To appear in Journal of Algebraic Combinatoric