Cyclic sieving, necklaces, and branching rules related to Thrall's problem

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 Kraśkiewicz--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.