  Moments of secular and inverse secular coefficients, averaged over random
matrices from classical groups, are related to the enumeration of non-negative
matrices with prescribed row and column sums. Similar random matrix averages
are related to certain configurations of vicious random walkers and to the
enumeration of plane partitions. The combinatorial meaning of the average of
the characteristic polynomial of random Hermitian and Wishart matrices is also
investigated, and consequently several simple universality results are derived.
