I'm a combinatorialist that is interested in estimating multiplicities of irreps of $1^{S_{kn}}_{S_k \wr S_n}$ (the action of symmetric group on uniform partitions). I'm aware of the difficulty (or perhaps not) and would be content with decent estimates for small $k$, since a good concrete estimate would have an application to an extremal uniform set partition problem. Let $f^\lambda$ be the dimension of the Specht module $S^\lambda$ and let $m_\lambda$ be its multiplicity in the irrep decomposition of $1^{S_{kn}}_{S_k \wr S_n}$. It's not hard to see that

$$ \sum_{\lambda \vdash kn} m_\lambda~f^\lambda = |S_{kn}/(S_{k} \wr S_n)|.$$

For $k = 2$ it is known that $0 \leq m_\lambda \leq 1$ and $m_\lambda = 1$ iff $\lambda \vdash 2n$ is of the form $2\mu := (2\mu_1,2\mu_2,\cdots,2\mu_s)$ for some $\mu = (\mu_1,\mu_2,\cdots,\mu_s) \vdash n$ (in other words, $1^{S_{2n}}_{S_2 \wr S_n}$ is multiplicity-free whose support are those shapes $2\mu$). Note that any $\lambda \vdash 2n$ with more than $n$ parts must occur with multiplicity zero. It seems to me that this should hold in general, that is, I'd like to know the following.

(Conjecture) Let $\lambda = (\lambda_1,\lambda_2,\cdots,\lambda_s) \vdash kn$. If $n < s$, then the corresponding Specht module $S^\lambda$ occurs with multiplicity zero in the irrep decomposition of $1^{S_{kn}}_{S_k \wr S_n}$.

If this claim is an exercise, then I'd prefer a couple of good hints over a proof sketch.

The conjecture is clearly true for $k = 2$ and I suspect that its true for $k = 3$ (I've verified this for very small $n$); however, I cannot be sure since to my knowledge no one has given a formula for determining multiplicities of the irreps of $1^{S_{3n}}_{S_3 \wr S_n}$. Thrall (1942) did this for $1^{S_{3n}}_{S_n \wr S_3}$, and since Foulkes' conjecture is true for $k \leq 4$, we have that the multiplicity of each irrep $\lambda$ of $1^{S_{3n}}_{S_3 \wr S_n}$ is at least that of $\lambda$ in $1^{S_{3n}}_{S_n \wr S_3}$.

Now if the above conjecture is true, then it linearly constrains the number of vectors $m \in \mathbb{N}^{\lambda(kn)}$ that meet the equation above with equality where $\lambda(kn)$ is the number of integer partitions of $kn$. Since Foulkes' conjecture is true for $k = 3$, the multiplicities of $1^{S_{3n}}_{S_n \wr S_3}$ (which are known) provide more linear constraints. It's probably too optimistic, but I'm curious if the solution set can be constrained enough along these lines to estimate or say something useful about the multiplicities of irreps of $1^{S_{3n}}_{S_3 \wr S_n}$.

I'm no algebraist, so any comments or pointers to germane papers would be greatly appreciated.