  We apply lattice point counting methods to compute the multiplicities in the
plethysm of $GL(n)$. Our approach gives insight into the asymptotic growth of
the plethysm and makes the problem amenable to computer algebra. We prove an
old conjecture of Howe on the leading term of plethysm. For any partition $\mu$
of 3,4, or 5 we obtain an explicit formula in $\lambda$ and $k$ for the
multiplicity of $S^\lambda$ in $S^\mu(S^k)$.
