On the Number of Matrices and a Random Matrix with Prescribed Row and Column Sums and 0-1 Entries
نویسنده
چکیده
We 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.
منابع مشابه
Matrices with Prescribed Row and Column Sums
This is a survey of the recent progress and open questions on the structure of the sets of 0-1 and non-negative integer matrices with prescribed row and column sums. We discuss cardinality estimates, the structure of a random matrix from the set, discrete versions of the Brunn-Minkowski inequality and the statistical dependence between row and column sums.
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