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International Mathematics Research Notices Advance Access originally published online on December 25, 2008
International Mathematics Research Notices (2009) 2009:348-385, doi:10.1093/imrn/rnn133 published on January 23, 2009
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© The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org.

Asymptotic Estimates for the Number of Contingency Tables, Integer Flows, and Volumes of Transportation Polytopes

Alexander Barvinok

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA

Correspondence: Correspondence to be sent to: barvinok{at}umich.edu

We prove an asymptotic estimate for the number of m x n non-negative integer matrices (contingency tables) with prescribed row and column sums and, more generally, for the number of integer-feasible flows in a network. Similarly, we estimate the volume of the polytope of m x n non-negative real matrices with prescribed row and column sums. Our estimates are solutions of convex optimization problems, and hence can be computed efficiently. As a corollary, we show that if row sums R = (r1, ..., rm) and column sums C = (c1, ..., cn) with r1 + ··· + rm = c1 + ··· + cn = Nare sufficiently far from constant vectors, then, asymptotically, in the uniform probability space of the m x nnon-negative integer matrices with the total sum N of entries, the event consisting of the matrices with row sums R and the event consisting of the matrices with column sums C are positively correlated.


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