3-Plethysms of homogeneous and elementary symmetric functions

We introduce the new combinatorial approach of plethystic type of tableaux, as a method to understand coefficients of Schur functions appearing in plethysms $s_{\nu }[h_{\lambda }]$ and $s_{\nu }[e_{\lambda }]$, for any partitions $\lambda$ and $\nu$. We first give general results about this approach, then use results on tableaux, ribbon tableaux and integer points in polytopes to understand the case where $\nu$ is a partition of $3$ and $\lambda$ has one part. We then use a $\mathit{\text{Kronecker}}\mathit{\text{map}}$ to extend these results to any partition $\lambda$.