An Extension of MacMahon's Equidistribution Theorem to Ordered Multiset Partitions

Andrew Timothy Wilson

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.

Keywords


Inversion number; Major index; Permutation statistics; Insertion method; Ordered multiset partitions; Macdonald polynomials

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