  Let the term $k$-representation refer to the permutation representations of
the symmetric group $\mathfrak{S}_n$ on $k$-tuples and $k$-subsets as well as
the $S^{(n-k,1^k)}$ irreducible representation of $\mathfrak{S}_n$. Endow
$\mathfrak{S}_n$ with the Ewens distribution and let $\alpha$ and $\beta$ be
linearly independent irrational numbers over $\mathbb{Q}$. Then for fixed $k >
1$ we show that as $n \to \infty$, the normalized count of the number of
eigenangles in a fixed interval $(\alpha, \beta)$ of a $k$-representation
evaluated at a random element $\sigma \in \mathfrak{S}_n$ converges weakly to a
compactly supported distribution. In particular, we compute the limiting
moments and moreover provide an explicit formula for the limiting density when
$k = 2$ and the Ewens parameter $\theta = 1$ (uniform probability measure).
This is in contrast to the $k = 1$ case where it has been shown previously that
the distribution is asymptotically Gaussian.
