It is well known that irreducible characters of the symmetric group satisfy orthogonality relations,
$$ \sum_{\mu \in P(n)} \chi_\mu^\lambda\chi_\mu^\omega=z_\lambda\delta_{\lambda,\omega},\quad \sum_{\lambda \in P(n)} \frac{1}{z_\lambda}\chi_\mu^\lambda\chi_\omega^\lambda=\delta_{\mu,\omega},$$
where $P(n)$ is the set of partitions of $n$ and $n!/z_\lambda$ is the size of a conjugacy class in that group.
I would like to know what has been proved in terms of partial or restricted such sums, i.e. quantities like $$ \sum_{\mu \in Q} \chi_\mu^\lambda\chi_\mu^\omega \quad \text{or}\quad \sum_{\lambda \in Q} \frac{1}{z_\lambda}\chi_\mu^\lambda\chi_\omega^\lambda,$$ where $Q\subset P(n)$.
