Some plethysm results related to Foulkes' conjecture (Q819186)

Summary: We provide several classes of examples to show that Stanley's ple\-thysm conjecture [\textit{P. R. Stanley}, in: Mathematics: frontiers and perspectives, 295--316 (2000; Zbl 0955.05111)] and a reformulation by \textit{P. Pylyavskyy} [Electron. J. Comb. 11, Research paper 8, electronic only (2004; Zbl 1060.05099)], 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; see \textit{S. C. Black} and \textit{R. J. List} [Eur. J. Comb. 10, 111--112 (1989; Zbl 0668.20017)]. 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.