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](http://www.combinatorics.org/ojs/index.php/eljc/article/view/v11i1r11), 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.



Is the reduced plethysm (restricted to 2-columns in Young tableaux) of this Schur funtion known $\mathbb S_{3^1}(\mathbb S_{1^p})$?
This follows on from Dan Peterson's answer. For $r,s,a \in \mathbb{N}\_0$, let $N\_{r,s}(a)$ be the number of partitions of $a$ whose Young diagram fits into a box with $r$ rows and $s$ columns. By the Cayley–Sylvester formula, the multiplicity of $s\_{(rs-a,a)}$ in $s\_{(r)} \circ s\_{(s)}$ is $N\_{r,s}(a) - N\_{r,s}(a-1)$ for $a$ such that $1 \le a \le rs/2$. See Corollary 2.12 in [this paper](http://arxiv.org/abs/1207.6300) of Giannelli for a short proof using the symmetric group, or [this paper](http://hal.inria.fr/docs/00/33/05/96/PDF/meulien.pdf) of Manivel for a generalization.


So the multiplicity of $s\_{(3p-a,a)}$ in $s\_{(3)} \circ s\_{(p)}$ is $N\_{(3,p)}(a) - N\_{(3,p)}(a-1)$ for $a \le 3p/2$.


When $p$ is odd we need instead the multiplicity of $s\_{(3p-a,a)}$ in $s\_{(1^3)} \circ s\_{(p)}$. By the Wronskian isomorphism (see [here](https://mathoverflow.net/questions/121753/known-decomposition-of-bigwedgek-symd-mathbb-cn-in-special-cases) or Manivel's paper), we have


$$ \langle s\_{(1^3)} \circ s\_{(p)}, s\_{(3p-a,a)} \rangle = \langle s\_{(3)} \circ s\_{(p-2)}, s\_{(3p-a-3,a-3)} \rangle = N\_{(3,p-2)}(a-3) - N\_{(3,p-2)}(a-4) $$


for $p \ge 2$ and $4 \le a \le 3p/2$. If $a \le 2$ then the multiplicity is zero and if $a=3$ the multiplicity is $1$.


