ON THE VOLUME OF THE POLYTOPE OF DOUBLY STOCHASTIC MATRICES

CLARA S. CHAN AND DAVID P. ROBBINS

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 decomposition 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.