  We make progress towards understanding the structure of Littlewood-Richardson
coefficients $g_{\lambda,\mu}^{\nu}$ for products of Jack symmetric functions.
Building on recent results of the second author, we are able to prove new cases
of a conjecture of Stanley in which certain families of these coefficients can
be expressed as a product of upper or lower hook lengths for every box in each
of the partitions. In particular, we prove that conjecture in the case of a
rectangular union, i.e. for $g_{\mu,\bar \sigma}^{\mu \cup m^n}$ where $\bar
\sigma$ is the complementary partition of $\sigma = \mu \cap m^n$ in the
rectangular partition $m^n$. We give a formula for these coefficients through
an explicit prescription of such choices of hooks. Lastly, we conjecture an
analogue of this conjecture of Stanley holds in the case of Shifted Jack
functions.
