Let $\lambda$ be a partition of $n$ and $\chi^\lambda$ be the character of $S_n$ associated to it. Given any conjugacy class $C$, I want to prove that $$\lambda\mapsto \frac{\chi^\lambda(C)}{f^\lambda}=\frac{\chi^\lambda(C)}{\chi^\lambda(\text{id})}=\frac{\chi^\lambda(C)}{\dim\chi^\lambda}$$ is a polynomial.
This was affirmed without proof in [1] (Proposition 2.9). The article refers to [2] for a proof. In [2] the proof is not straightforward at all (at least for a undergraduate). I think that, since we want to prove something weaker it should be easier.
[1] Eskin, Alex, and Andrei Okounkov. "Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials." Inventiones Mathematicae 145.1 (2001): 59-103.
[2] Kerov, Serguei. "Polynomial functions on the set of Young diagrams." CR Acad. Sci. Paris Sér. I Math. 319 (1994): 121-126.