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

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. R´esum´e. Un r´esultat classique de MacMahon affirme que nombre d’inversion et l’indice majeur ont la mˆeme distribution sur permutations d’un multi-ensemble donn´e. Dans ce travail, nous d´emontrons un renforcement de ce th´eor`eme origine conjectur´e par Haglund. Notre r´esultat peut ˆetre consid´er´e comme un th´eor`eme d’´equir´epartition sur l...

• Type: Journal
• Year: 2016
• Where: COMBINATORICS
• Authors: Andrew Timothy Wilson
• Keywords: COMBINATORICS 2016