  This paper is devoted to the distribution of the eigenvalues of $XUYU^t$
where $X$ and $Y$ are given symmetric matrices and $U$ is a random real valued
square matrix of standard normal distribution. More specifically we look at its
moments, i.e. the mathematical expectation of the trace of $(XUYU^t)^n$ for
arbitrary integer $n$. Hanlon, Stanley, Stembridge (1992) showed that this
quantity can be expressed in terms of some generating series for the connection
coefficients of the double cosets of the hyperoctahedral group with the
eigenvalues of $X$ and $Y$ as indeterminate. We provide an explicit evaluation
of these series in terms of monomial symmetric functions. Our development
relies on an interpretation of the connection coefficients in terms of locally
orientable hypermaps and a new bijective construction between partitioned
locally orientable hypermaps and some decorated forests. As a corollary we
provide a simple explicit evaluation of the moments of $XUYU^*$ when $U$ is
complex valued and $X$ and $Y$ are given hermitian matrices.
