  Representation theory of the symmetric group $\mathfrak{S}_n$ has a very
distinctive combinatorial flavor. The conjugacy classes as well as the
irreducible characters are indexed by integer partitions $\lambda \vdash n$. We
introduce class functions on $\mathfrak{S}_n$ that count the number of certain
tilings of Young diagrams. The counting interpretation gives a uniform
expression of these class functions in the ring of character polynomials, as
defined by \cite{murnaghanfirst}. A modern treatment of character polynomials
is given in \cite{orellana-zabrocki}. We prove a relation between these
combinatorial class functions in the (virtual) character ring. From this
relation, we were able to prove Goupil's generating function identity
\cite{goupil}, which can then be used to derive Rosas' formula \cite{rosas} for
Kronecker coefficients of hook shape partitions and two row partitions.
