  This paper proves a combinatorial rule giving all maximal and minimal
partitions $\lambda$ such that the Schur function $s_\lambda$ appears in a
plethysm of two arbitrary Schur functions. Determining the decomposition of
these plethysms has been identified by Stanley as a key open problem in
algebraic combinatorics. As corollaries we prove three conjectures of Agaoka on
the partitions labelling the lexicographically greatest and least Schur
functions appearing in an arbitrary plethysm. We also show that the
multiplicity of the Schur function labelled by the lexicographically least
constituent may be arbitrarily large. The proof is carried out in the symmetric
group and gives an explicit non-zero homomorphism corresponding to each maximal
or minimal partition.
