Peak quasisymmetric functions and Eulerian enumeration

By Louis J. Billera, Samuel K. Hsiao, Stephanie van Willigenburg. 2003

78 citations recorded by the source corpus.

Abstract Via duality of Hopf algebras, there is a direct association between peak quasisymmetric functions and enumeration of chains in Eulerian posets. We study this association explicitly, showing that the notion of cd -index, long studied in the context of convex polytopes and Eulerian posets, arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by Stembridge. Thus Eulerian posets having a nonnegative cd -index (for example, face lattices of convex polytopes) correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis. We diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions, and study the g -polynomial for Eulerian posets as an algebra homomorphism.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/e5affdafd781762a430f032455ea516af5968acc

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

The peak algebra and the descent algebras of types B and D (2003, background): “More recent works uncovered connections between peaks and such disparate topics as the generalized Dehn-Sommerville equations [6, 10, 1] and the Schubert calculus of isotropic (cid:13)ag manifolds [11, 8, 7].”
Enriched P-partitions and peak algebras (2005, methodology): “The study of algebras related to peaks began with John Stembridge’s paper [30] on enriched P-partitions, followed by others, including [1, 2, 7, 8, 9 , 21, 27].”