Hall-Littlewood polynomials and fixed point enumeration

TL;DR

This paper proves conjectures on counting transportation matrices invariant under cyclic rotations, using symmetric functions like Hall-Littlewood polynomials and the bicyclic sieving phenomenon.

Contribution

It provides an affirmative resolution to conjectures involving cyclic symmetry in transportation matrices using advanced symmetric function techniques.

Findings

Confirmed conjectures on cyclically invariant transportation matrices

Connected enumeration problems to the bicyclic sieving phenomenon

Applied Hall-Littlewood polynomials and rim hook correspondence

Abstract

We resolve affirmatively some conjectures of Reiner, Stanton, and White \cite{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 \cite{BRSBiD}. The proofs of our results use various tools from symmetric function theory such as the Stanton-White rim hook correspondence \cite{SW} and results concerning the specialization of Hall-Littlewood polynomials due to Lascoux, Leclerc, and Thibon \cite{LLTUnity} \cite{LLTRibbon}.