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 x n doubly stochastic matrices (real nonnegative matrices with 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", that is, n x n nonnegative 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.