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We establish an equidistribution between the pairs (inv, maj) and (quinv, maj) on any row-equivalency class where is a filling of a given Young diagram. In particular if is a filling of a rectangular diagram, the triples (inv, quinv, maj) and (quinv, inv, maj) have the same distribution over. Our main result affirms a conjecture proposed by Ayyer, Mandelshtam and Martin (2021), thus presenting the equivalence between two refined formulas for the modified Macdonald polynomials.
Jin et al. (Fri,) studied this question.