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

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. As an application, we develop refined Macdonald polynomials for hook shapes. We show that these polynomials are symmetric and give their Schur expansion.

Keywords

• inversion number
• major index
• permutation statistics
• insertion method
• ordered multiset partitions
• macdonald polynomials
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]
• [math.math-co] mathematics [math]/combinatorics [math.co]