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{\bf Steven Sivek}
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{\bf Some Plethysm Results related to Foulkes' Conjecture}
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We provide several classes of examples to show that Stanley's ple\-thysm
conjecture and a reformulation by Pylyavskyy, both concerning the
ranks of certain matrices $K^{\lambda}$ associated with Young diagrams
$\lambda$, are in general false. We also provide bounds on the rank
of $K^{\lambda}$ by which it may be possible to show that the approach
of Black and List to Foulkes' conjecture does not work in general.
Finally, since Black and List's work concerns $K^{\lambda}$ for rectangular
shapes $\lambda$, we suggest a constructive way to prove that $K^{\lambda}$
does not have full rank when $\lambda$ is a large rectangle.
\bye