A poset view of the major index

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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 languageEnglish
Pages (from-to)1-14
Number of pages14
JournalAdvances in Applied Mathematics
Volume62
DOIs
StatePublished - Jan 1 2015

Bibliographical note

Publisher Copyright:
© 2014 Elsevier Inc. Allrightsreserved.

Funding

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).

FundersFunder number
Simons Foundation206001
National Security AgencyH98230-13-1-0280

    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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