Macdonald symmetry at q=1 and a new class of inv-preserving bijections on words

We give a direct combinatorial proof of the q,t-symmetry relation H̃_μ(X;q,t)=H̃_μ'(X;t,q) in the Macdonald polynomials H̃_μ at the specialization q=1. The bijection demonstrates that the Macdonald inv statistic on the permutations of any given row of a Young diagram filling is Mahonian. Moreover, our bijection gives rise a family of new bijections on words that preserves the classical Mahonian inv statistic.