Counting binary matrices with given row and column sums

Integral Matrices with Given Row and Column Sums

Let P = (p,,) and Q = (qij) be m x n integral matrices, R and S be integral vectors. Let Nf(R, S) denote the class of all m x n integral matrices A with row sum vector R and column sum vector S satisfying P < A < Q. For a wide variety of classes ‘%$I( R, S) satisfying our main condition, we obtain two necessary and sufficient conditions for the existence of a matrix in @(R, 5). The first charac...

On nonnegative matrices with given row and column sums

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.

Asymptotic Enumeration of Binary Matrices with Bounded Row and Column Sums

Let An be the set of all n×n binary matrices in which the number of 1’s in each row and column is at most n/2. We show that |An| = 2n−ρn+δ √ n · nO(1), for a constant ρ ≈ 1.42515, and δ = δ(n) ≈ 1.46016 for even n and 0 otherwise.

Subdominant eigenvalues for stochastic matrices with given column sums

For any stochastic matrix A of order n, denote its eigenvalues as λ1(A), . . . , λn(A), ordered so that 1 = |λ1(A)| ≥ |λ2(A)| ≥ . . . ≥ |λn(A)|. Let cT be a row vector of order n whose entries are nonnegative numbers that sum to n. Define S(c), to be the set of n × n row-stochastic matrices with column sum vector cT . In this paper the quantity λ(c) = max{|λ2(A)||A ∈ S(c)} is considered. The ve...