  We represent the number of mxn non-negative integer matrices (contingency
tables) with prescribed row sums and column sums as the expected value of the
permanent of a non-negative random matrix with exponentially distributed
entries. We bound the variance of the obtained estimator, from which it follows
that if the row and column sums are bounded by a constant fixed in advance, we
get a polynomial time approximation scheme for counting contingency tables. We
show that the complete symmetric polynomial of a fixed degree in n variables
can be epsilon-approximated coefficient-wise by a sum of powers of O(log n)
linear forms, from which it follows that if the row sums (but not necessarily
column sums) are bounded by a constant, there is a deterministic approximation
algorithm of m^{O(log n)} complexity to compute the logarithmic asymptotic of
the number of tables.
