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Séminaire Lotharingien de Combinatoire, 82B.37 (2019), 12 pp.

# Connor Ahlbach and Joshua P. Swanson

# Thrall's problem: cyclic sieving, necklaces, and branching rules

**Abstract.**
The cyclic sieving phenomenon (CSP) of Reiner-Stanton-White offers an explanation for why many polynomials evaluate to non-negative integers at roots of unity. We reverse the usual paradigm and show how certain cyclic sieving results can be used to give new proofs of Schur expansions due to Kra\'skiewicz-Weyman, Stembridge, and Schocker. These results concern the so-called higher Lie modules and branching rules for inclusions *C*_{a} wreath *S*_{b} -> *S*_{ab}. Extending the approach gives monomial expansions for certain Frobenius series arising from a generalization of Thrall's problem.

Received: November 15, 2018.
Accepted: February 17, 2019.
Final version: April 1, 2019.

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