On the eigenvalues multiplicity function of the Star graph

The Star graph is the Cayley graph on the symmetric group $\mathrm{Sym}_n$ generated by the set of transpositions $\{(1 2),(1 3),\ldots,(1 n)\}$. We consider the spectrum of the Star graph as the spectrum of its adjacency matrix. The spectrum of $S_n$ is integral as it was shown independently by R. Krakovski, B. Mohar, and G. Chapuy, V. Feray in 2012. In this paper we show that the multiplicity of eigenvalues of the Star graph is a polynomial in the indeterminate $n$ of degree $2(t-1)$ with leading coefficient $\frac{1}{(t-1)!}$.