An extension of MacMahon's Equidistribution Theorem to ordered multiset partitions

TL;DR

This paper extends MacMahon's classical equidistribution theorem to ordered multiset partitions, introduces a bijective proof involving a generalized insertion method, and applies it to extend Macdonald polynomials for hook shapes, proving their symmetry and Schur expansion.

Contribution

It proves a strengthened equidistribution theorem for ordered multiset partitions, generalizes Carlitz's insertion method, and extends Macdonald polynomials to new symmetric forms.

Findings

Inversion number and major index are equidistributed over ordered multiset partitions.

A new bijective proof using a generalized insertion method is provided.

Extended Macdonald polynomials for hook shapes are shown to be symmetric with explicit Schur expansion.

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 this 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.