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

Andrew Timothy Wilson

doi:10.46298/dmtcs.2405

Andrew Timothy Wilson - 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) - https://doi.org/10.46298/dmtcs.2405

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

Authors: Andrew Timothy Wilson ¹

1 University of California [San Diego] [UC San Diego]

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.

https://doi.org/10.46298/dmtcs.2405

Source: 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

Imported 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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