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. 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.Comment: An extended abstract of this work was presented at FPSAC 201

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

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.Un résultat classique de MacMahon affirme que nombre d’inversion et l’indice majeur ont la même distribution sur permutations d’un multi-ensemble donné. Dans ce travail, nous démontrons un renforcement de ce théorème origine conjecturé par Haglund. Notre résultat peut être considéré comme un théorème d’équirépartition sur les partitions ordonnées d’un multi-ensemble en ensembles, que nous appellerons partitions de multiset commandés. Notre preuve est bijective et implique une nouvelle généralisation de la méthode d’insertion de Carlitz. Comme application, nous développons des polynômes de Macdonald raffinés pour formes d’hameçons. Nous montrons que ces polynômes sont symétriques et donnent leur expansion Schur