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

Abstract

We consider the set Σ ( R , C ) of all m × n matrices having 0–1 entries and prescribed row sums R = ( r 1 , … , r m ) and column sums C = ( c 1 , … , c n ) . 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.