  We continue our study, initiated in our prior work with Richard Stanley, of
the representation of the symmetric group on the multilinear component of an
$n$-ary generalization of the free Lie algebra known as the free Fillipov
$n$-algebra with $k$ brackets. Our ultimate aim is to determine the
multiplicities of the irreducible representations in this representation. This
had been done for the ordinary Lie representation ($n=2$ case) by Kraskiewicz
and Weyman. The $k=2$ case was handled in our prior work, where the
representation was shown to be isomorphic to $S^{2^{n-1}1}$. In this paper, for
general $n$ and $k$, we obtain decomposition results that enable us to prove
that in the $k=3$ case, the representation is isomorphic to $S^{3^{n-1}1}
\oplus S^{3^{n-2}21^2}$.
