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

Abstract

We give a direct combinatorial proof of the $q,t$-symmetry relation ${\tilde{H}}_{\mu }(X;q,t)={\tilde{H}}_{{\mu }^{\prime }}(X;t,q)$ in the Macdonald polynomials ${\tilde{H}}_{\mu }$ 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.

10 pages; submitted to FPSAC 2017