The Tchebyshev transforms of the first and second kind

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12 Scopus citations


An in-depth study of the Tchebyshev transforms of the first and second kind of a poset is taken. The Tchebyshev transform of the first kind is shown to preserve desirable combinatorial properties, including EL-shellability and nonnegativity of the cd-index. When restricted to Eulerian posets, it corresponds to the Billera, Ehrenborg, and Readdy omega map of oriented matroids. The Tchebyshev transform of the second kind U is a Hopf algebra endomorphism on the space of quasisymmetric functions which, when restricted to Eulerian posets, coincides with Stembridge's peak enumerator. The complete spectrum of U is determined, generalizing the work of Billera, Hsiao, and van Willigenburg. The type B quasisymmetric function of a poset is introduced and, like Ehrenborg's classical quasisymmetric function of a poset, it is a comodule morphism with respect to the quasisymmetric functions QSym. Finally, similarities among the omega map, Ehrenborg's r-signed Birkhoff transform, and the Tchebyshev transforms motivate a general study of chain maps which occur naturally in the setting of combinatorial Hopf algebras.

Original languageEnglish
Pages (from-to)211-244
Number of pages34
JournalAnnals of Combinatorics
Issue number2
StatePublished - Jun 2010

Bibliographical note

Funding Information:
Acknowledgments. We graciously thank Gábor Hetyei for inspiring us to study the Tcheby-shev transform and suggesting research directions. We thank Ira Gessel for directing us to Chak-On Chow’s work on the type B quasisymmetric functions. The authors also thank the referees for their comments and suggestions. The first author was partially supported by National Science Foundation grant 0200624 and by a University of Kentucky College of Arts & Sciences Faculty Research Fellowship. The second author was partially supported by a University of Kentucky College of Arts & Sciences Research Grant. Both authors thank the Institute for Advanced Study/Park City Mathematics Institute for providing a stimulating work environment where this work was initiated and MIT, where the authors finished this work while on sabbatical.


  • Eulerian posets
  • Hopf algebra
  • Poset transforms
  • Quasisymmetric functions
  • cd-index

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics


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