Let $\lambda$, $\mu$, and $\rho$ be partitions of $n$ and let $\chi^\lambda_\rho$ and $K_{\lambda \mu}$ denote the associated ${\frak{S}}_n$-character value and Kostka number respectively.
Question: Is there a good combinatorial and/or representation-theoretic interpretation of the quantity
\begin{equation} \sum_{\lambda \vdash n} \, \chi^\lambda_\rho K_{\lambda \mu} \end{equation}
What happens if I replace $K_{\lambda \mu}$ by the Kostka-Foulkes polynomial $K_{\lambda \mu}(t)$ ?
thanks, ines.