  The integrable structure of Ginibre's Orthogonal Ensemble of random matrices
is looked at through the prism of the probability "p_{n,k}" to find exactly "k"
real eigenvalues in the spectrum of an "n" by "n" real asymmetric Gaussian
random matrix. The exact solution for the probability function "p_{n,k}" is
presented, and its remarkable connection to the theory of symmetric functions
is revealed. An extension of the Dyson integration theorem is a key ingredient
of the theory presented.
