Ramanujan sums and rectangular power sums

For a fixed nonnegative integer $u$ and positive integer $n$, we investigate the symmetric function $$\sum _{d\mathrm{\mid }n}{(c_d(\frac{n}{d}))}^u{p}_d^{\frac{n}{d}},$$ where $p_n$ denotes the $n$th power sum symmetric function, and $c_d(r)$ is a Ramanujan sum, equal to the sum of the $r$th powers of all the primitive $d$th roots of unity. We establish the Schur positivity of these functions for $u=0$ and $u=1$, showing that, in each case, the associated representation of the symmetric group ${\mathfrak{S}}_n$ decomposes into a sum of Foulkes representations, that is, representations induced from the irreducibles of the cyclic subgroup generated by the long cycle. We also conjecture Schur positivity for the case $u=2$.