A poset view of the major index

Richard Ehrenborg; Margaret Readdy

doi:10.1016/j.aam.2014.09.005

A poset view of the major index

Richard Ehrenborg, Margaret Readdy

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We introduce the Major MacMahon map from ℤ<a, b> to Z[q], and show how this map interacts with the pyramid and bipyramid operators. When the Major MacMahon map is applied to the ab-index of a simplicial poset, it yields the q-analogue of n! times the h-polynomial of the poset. Applying the map to the Boolean algebra gives the distribution of the major index on the symmetric group, a seminal result due to MacMahon. Similarly, when applied to the cross-polytope we obtain the distribution of one of the major indexes on signed permutations due to Reiner.

Original language	English
Pages (from-to)	1-14
Number of pages	14
Journal	Advances in Applied Mathematics
Volume	62
DOIs	https://doi.org/10.1016/j.aam.2014.09.005
State	Published - Jan 1 2015

Bibliographical note

Funding Information:
The authors thank the referee for his careful comments. The first author was partially supported by National Security Agency grant H98230-13-1-0280 . This work was partially supported by a grant from the Simons Foundation (# 206001 to Margaret Readdy).

Publisher Copyright:
© 2014 Elsevier Inc. Allrightsreserved.

Keywords

Permutations and signed permutations
Principal specialization
Simplicial posets
The boolean algebra and the face lattice of a cross-polytope
The major index

ASJC Scopus subject areas

Applied Mathematics

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10.1016/j.aam.2014.09.005

Cite this

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abstract = "We introduce the Major MacMahon map from ℤ to Z[q], and show how this map interacts with the pyramid and bipyramid operators. When the Major MacMahon map is applied to the ab-index of a simplicial poset, it yields the q-analogue of n! times the h-polynomial of the poset. Applying the map to the Boolean algebra gives the distribution of the major index on the symmetric group, a seminal result due to MacMahon. Similarly, when applied to the cross-polytope we obtain the distribution of one of the major indexes on signed permutations due to Reiner.",

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AB - We introduce the Major MacMahon map from ℤ to Z[q], and show how this map interacts with the pyramid and bipyramid operators. When the Major MacMahon map is applied to the ab-index of a simplicial poset, it yields the q-analogue of n! times the h-polynomial of the poset. Applying the map to the Boolean algebra gives the distribution of the major index on the symmetric group, a seminal result due to MacMahon. Similarly, when applied to the cross-polytope we obtain the distribution of one of the major indexes on signed permutations due to Reiner.

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A poset view of the major index

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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