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

TL;DR

The paper provides a combinatorial proof of Macdonald polynomial symmetry at q=1 and introduces new bijections on words that preserve the Mahonian inv statistic, revealing deeper combinatorial structures.

Contribution

It offers a direct combinatorial proof of Macdonald symmetry at q=1 and constructs new inv-preserving bijections on words, connecting Macdonald statistics with Mahonian properties.

Findings

Proves Macdonald symmetry combinatorially at q=1

Constructs new bijections preserving Mahonian inv statistic

Shows inv statistic on permutations is Mahonian

Abstract

We give a direct combinatorial proof of the $q,t$-symmetry relation $\tilde H_{\mu}(X;q,t)=\tilde H_{\mu'}(X;t,q)$ 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.