Polynomial Expressions for Symmetric Group Characters on Cycles

In \cite{[CZ]}, Cohen and Zemel showed that for a partition $\lambda \vdash k$, the dimension of the irreducible representation of $S_{n}$ corresponding to the partition $(n-k,\lambda) \vdash n$ is a polynomial of degree $k$ in $n$, whose coefficients in the binomial basis count standard Young tableaux of shape $\lambda$ with special restrictions. In this paper, we generalize their results on the representation's dimension to character values on arbitrary cycles.