The r-signed Birkhoff transform

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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 ωr. We also obtain new expressions for the ab-index of the r-cubical lattice in terms of the map ωr applied to all the permutations in the symmetric group. We show that the map r⋅ωrr is an endomorphism on the Hopf algebra of quasisymmetric functions.

Original languageEnglish
Article number112214
JournalDiscrete Mathematics
Issue number2
StatePublished - Feb 2021

Bibliographical note

Publisher Copyright:
© 2020 Elsevier B.V.


  • Birkhoff transform
  • r-cubical lattice

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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