On the Volume of the Polytope of Doubly Stochastic Matrices

We study the calculation of the volume of the polytope Bn of n × n doubly stochastic matrices; that is, the set of real non-negative matrices with all row and column sums equal to one. We describe two methods. The first involves a decomposition of the polytope into simplices. The second involves the enumeration of “magic squares”, i.e., n×n non-negative integer matrices whose rows and columns all sum to the same integer. We have used the first method to confirm the previously known values through n = 7. This method can also be used to compute the volumes of faces of Bn. For example we have observed that the volume of a particular face of Bn appears to be a product of Catalan numbers. We have used the second method to find the volume for n = 8, which we believe was not previously known.