On the volume of the polytope of doubly stochastic matrices

Abstract

We study the calculation of the volume of the polytope B_n of n by 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 by 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 B_n. For example, we have observed that the volume of a particular face of B_n 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.