Wilson, Andrew Timothy - An extension of MacMahon's Equidistribution Theorem to ordered multiset partitions

dmtcs:2405 - Discrete Mathematics & Theoretical Computer Science, January 1, 2014, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)
An extension of MacMahon's Equidistribution Theorem to ordered multiset partitions

Authors: Wilson, Andrew Timothy

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.


Source : oai:HAL:hal-01207606v1
Volume: DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)
Section: Proceedings
Published on: January 1, 2014
Submitted on: November 21, 2016
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]


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