We establish an equidistribution between the pairs (inv,maj) and (quinv,maj) on any row-equivalency class [τ] where τ is a filling of a given Young diagram. In particular if τ is a filling of a rectangular diagram, the triples (inv,quinv,maj) and (quinv,inv,maj) have the same distribution over [τ]. 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.
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Jin et al. (2024) studied this question.
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