Abstract
A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work, we prove a strengthening of MacMahon's theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof is bijective and involves a new generalization of Carlitz's insertion method. This generalization leads to a new extension of Macdonald polynomials for hook shapes. We use our main theorem to show that these polynomials are symmetric and we give their Schur expansion.
 
 A corrigendum was added 17 September 2019.
Highlights
Given a composition α of length n, we let Sα be the set of all permutations of the multiset {iαi : 1 i n}
A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset
Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions
Summary
Given a composition α of length n (i.e. a vector of positive integers of length n), we let Sα be the set of all permutations of the multiset {iαi : 1 i n}. These sets allow us to define several statistics on Sα: des(σ) = | Des(σ)| asc(σ) = | Asc(σ)| inv(σ) = | Inv(σ)| maj(σ) =. These statistics are known as the descent number, ascent number, inversion number, and major index of σ, respectively. We show that these polynomials are symmetric and we give their expansion into the Schur function basis
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