On plethysm conjectures of Stanley and Foulkes: the $2\times n$ case

We prove Stanley’s plethysm conjecture for the $2\times n$ case, which composed with the work of Black and List provides another proof of Foulkes conjecture for the $2\times n$ case. We also show that the way Stanley formulated his conjecture, it is false in general, and suggest an alternative formulation.