  We show that the cyclic sieving phenomenon of Reiner--Stanton--White together
with necklace generating functions arising from work of Klyachko offer a
remarkably unified, direct, and largely bijective approach to a series of
results due to Kraskiewicz--Weyman, Stembridge, and Schocker related to the
so-called higher Lie modules and branching rules for inclusions $ C_a \wr S_b
\hookrightarrow S_{ab} $. Extending the approach gives monomial expansions for
certain graded Frobenius series arising from a generalization of Thrall's
problem.
