Asymptotic enumeration of integer matrices

Asymptotic enumeration of integer matrices.
                   B. D. McKay and J. C. McLeod
     Australian National University; University of Bristol
                                         Abstract
    Let s = (s1 , s2 , . . . , sm ) and t = (t1 , t2 , . . . , tn ) be vectors of non-negative
integers. Let M (m, s ; n, t ) be the number of m × n matrices over {0, 1, 2, . . . }
with the ith row summing to si and the j th column summing to tj . We are
interested in determining the asymptotic value of M (m, s ; n, t ) as m, n → ∞
under suitable conditions on s and t. In this talk we survey the work that has
been done in the area and explore some of the techniques used. In particular,
we will present the calculation for estimating the number of n × n symmetric
matrices over {0, 1, 2, . . . } with zeros on the main diagonal and each row and
column summing to , for sufficiently large .