| Summary: | <p> Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation <i>H˜</i>μ*(x; q,t) = <i>H˜</i><sub> μ</sub>(<i>x; t,q</i>). We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (<i>q</i> = 0) when mu is a partition with at most three rows, and for the coefficients of the square-free monomials in X={x_1,x_2,...} for all shapes mu. We also provide a proof for the full relation in the case when mu is a hook shape, and for all shapes at the specialization <i>t</i> = 1. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.</p>
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