On a certain expansion in term of Schur functions

Question

This question is related to this other one

A Schur positivity conjecture related to row and column permutations

by Richard Stanley (thanks to Sam Hopkins for letting me know about it).

Consider a Young subgroup $S_{\lambda}$ of the symmetric group $S_n$, corresponding to some integer partition $\lambda$ of $n$. Let $\tau$ be some permutation and define the symmetric function

$$ F(\tau)=\sum_{\sigma\in S_{\lambda}}p_{c(\tau\sigma)} $$ where $p_{\mu}$ is the usual power sum symmetric function and $c(\rho)$ denotes the integer partition given by the cycle-type of the permutation $\rho$.

Q: What is known about the Schur function expansion of $F(\tau)$, given the double coset class of $\tau$ for the Young subgroup?

Answer 1

One fact is that $F(\tau)$ is Schur positive if and only if $\tau\in S_\lambda$. More generally, if $K$ is any coset (left or right) of any subgroup $G$ of $S_n$, then $\sum_{\sigma\in K}p_{c(\sigma)}$ is Schur positive if and only if $K=G$. The only known proof for the "if" part requires representation theory; see Enumerative Combinatorics, vol. 2, page 396. For the "only if" part, it is easy to see that if a nonzero linear combination $\sum_{\lambda\vdash n} a_\lambda p_\lambda$ of power sums is Schur positive, then $a_{1^n}>0$.