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 × n doubly stochastic matrices (real nonnegative matrices with row and column sums equal to one). We describe two methods. The first involves a decompos ition of the polytope into simplices. The second involves the enumeration of “magic squares”, that is, n × 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.