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{\bf E. Rodney Canfield and Brendan D. McKay}
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{\bf Asymptotic Enumeration of Dense 0-1 Matrices with Equal Row Sums and Equal Column Sums}
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Let $s,\,t,\,m$ and $n$ be positive integers such that $sm=tn$.  Let
$B(m,s;n,t)$ be the number of $m\times n$ matrices over $\{0,1\}$ with
each row summing to $s$ and each column summing to~$t$.  Equivalently,
$B(m,s;n,t)$ is the number of semiregular bipartite graphs with $m$
vertices of degree $s$ and $n$ vertices of degree~$t$.  Define the
density $\lambda=s/n=t/m$.  The asymptotic value of $B(m,s;n,t)$ has been
much studied but the results are incomplete.  McKay and Wang (2003)
solved the sparse case $\lambda(1-\lambda)=o\big((mn)^{-1/2}\big)$ using
combinatorial methods.  In this paper, we use analytic methods to
solve the problem for two additional ranges.  In one range the matrix
is relatively square and the density is not too close to~0 or~1.  In
the other range, the matrix is far from square and the density is
arbitrary.  Interestingly, the asymptotic value of $B(m,s;n,t)$ can be
expressed by the same formula in all cases where it is known.  Based
on computation of the exact values for all $m,n\le 30$, we conjecture
that the same formula holds whenever $m+n\to\infty$ regardless of the
density.

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