Hall-Littlewood polynomials and fixed point enumeration
Abstract
We resolve affirmatively some conjectures of Reiner, Stanton, and White ReinerComm regarding enumeration of transportation matrices which are invariant under certain cyclic row and column rotations. Our results are phrased in terms of the bicyclic sieving phenomenon introduced by Barcelo, Reiner, and Stanton BRSBiD. The proofs of our results use various tools from symmetric function theory such as the Stanton-White rim hook correspondence SW and results concerning the specialization of Hall-Littlewood polynomials due to Lascoux, Leclerc, and Thibon LLTUnity LLTRibbon.
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