The r-signed Birkhoff transform

Richard Ehrenborg

doi:10.1016/j.disc.2020.112214

The r-signed Birkhoff transform

Richard Ehrenborg

Mathematics

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

Abstract

We introduce the r-signed Birkhoff transform of a distributive lattice, extending Hsiao's notion of the signed Birkhoff transform. We show how to compute the ab-index of the r-signed Birkhoff transform from the ab-index of the distributive lattice, generalizing work of Billera, Ehrenborg and Readdy, by extending their ω map to ω_r. We also obtain new expressions for the ab-index of the r-cubical lattice in terms of the map ω_r applied to all the permutations in the symmetric group. We show that the map r⋅ω_r=ϑ_r is an endomorphism on the Hopf algebra of quasisymmetric functions.

Original language	English
Article number	112214
Journal	Discrete Mathematics
Volume	344
Issue number	2
DOIs	https://doi.org/10.1016/j.disc.2020.112214
State	Published - Feb 2021

Bibliographical note

Funding Information:
The author thanks the referee for comments that lead to an improved exposition in this paper. The author thanks Margaret Readdy and Vic Reiner for discussions about complex polytopes. The author also thanks Margaret Readdy for helpful comments on an earlier version of this paper. The author also thanks the Institute for Advanced Study for a research visit 2019. The author was partially supported by National Science Foundation, United States of America grant 0200624 . This work was partially supported by a grant from the Simons Foundation, United States of America ( #429370 to Richard Ehrenborg).

Funding Information:
The author thanks the referee for comments that lead to an improved exposition in this paper. The author thanks Margaret Readdy and Vic Reiner for discussions about complex polytopes. The author also thanks Margaret Readdy for helpful comments on an earlier version of this paper. The author also thanks the Institute for Advanced Study for a research visit 2019. The author was partially supported by National Science Foundation, United States of America grant 0200624. This work was partially supported by a grant from the Simons Foundation, United States of America (#429370 to Richard Ehrenborg).

Publisher Copyright:
© 2020 Elsevier B.V.

Keywords

Birkhoff transform
r-cubical lattice

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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10.1016/j.disc.2020.112214

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The r-signed Birkhoff transform

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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