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1999 On the volume of the polytope of doubly stochastic matrices
Clara S. Chan, David P. Robbins
Experiment. Math. 8(3): 291-300 (1999).


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.


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Clara S. Chan. David P. Robbins. "On the volume of the polytope of doubly stochastic matrices." Experiment. Math. 8 (3) 291 - 300, 1999.


Published: 1999
First available in Project Euclid: 9 March 2003

zbMATH: 0951.05015
MathSciNet: MR1724161

Primary: 15A51
Secondary: 52A38

Rights: Copyright © 1999 A K Peters, Ltd.


Vol.8 • No. 3 • 1999
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