I am working on a physical problem, where I need to compute the "reduced plethysm" that is all the irreducibles characterised by the Young tableaux of 2 columns or less. The plethysm problem I want to solve is the following,

$$\mathbb s_{(3)}[\mathbb s_{1^p}] \text{ which can be written as } s_{(3)}[e_p],$$
where $e_p$ is the elementary symmetric polynomial of degree $p$,
and we work in the **permutation group $S_n$**, so everything is in $n$ variables.

**Edit:** Since your partitions are special cases of "hooks", you can use this reference, which gives an expansion in terms of Schur functions. The coefficients are given as certain character values, and are, according to the paper, difficult to compute in general. However, in your special case, it might be easier.