A Combinatorial Approach to the $q, t$-Symmetry Relation in Macdonald Polynomials

Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation H̃μ(x; q, t) = H̃μ∗(x; t, q). We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q = 0) when μ is a partition with at most three rows, and for the coefficients of the square-free monomials in x for all shapes μ. We also provide a proof for the full relation in the case when μ is a hook shape, and for all shapes at the specialization t = 1. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.