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On the number of matrices and a random matrix with prescribed row and column sums and 0–1 entries

By Alexander Barvinok

Abstract

AbstractWe consider the set Σ(R,C) of all m×n matrices having 0–1 entries and prescribed row sums R=(r1,…,rm) and column sums C=(c1,…,cn). We prove an asymptotic estimate for the cardinality |Σ(R,C)| via the solution to a convex optimization problem. We show that if Σ(R,C) is sufficiently large, then a random matrix D∈Σ(R,C) sampled from the uniform probability measure in Σ(R,C) with high probability is close to a particular matrix Z=Z(R,C) that maximizes the sum of entropies of entries among all matrices with row sums R, column sums C and entries between 0 and 1. Similar results are obtained for 0–1 matrices with prescribed row and column sums and assigned zeros in some positions

Publisher: Elsevier Inc.
Year: 2010
DOI identifier: 10.1016/j.aim.2009.12.001
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