On the number of matrices and a random matrix with prescribed row and column sums and 0-1 entries
Abstract
We consider the set Sigma(R,C) of all mxn 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 |Sigma(R, C)| via the solution to a convex optimization problem. We show that if Sigma(R, C) is sufficiently large, then a random matrix D in Sigma(R, C) sampled from the uniform probability measure in Sigma(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.
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