Hall–Littlewood polynomials and fixed point enumeration

We resolve affirmatively some conjectures of Reiner, Stanton, and White (2004) [12] 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 (2009) [1]. The proofs of our results use various tools from symmetric function theory such as the Stanton–White rim hook correspondence (Stanton and White (1985) [18]) and results concerning the specialization of Hall–Littlewood polynomials due to Lascoux, Leclerc, and Thibon (1994, 1997) [5] and [6].