A universal approach to quasisymmetric generating functions

Marcelo Aguiar

CRM-ISM, Université de Montréal

May 4,
refreshments at 3:45pm


We will discuss a universal procedure for the construction and study of several generating functions arising in combinatorics, within the framework of Hopf algebras. For several types of combinatorial objects, a natural Hopf algebra will be defined, and a "zeta" functional on it. To these, one associates a "Mobius" functional, a "zeta" polynomial and a "flag" quasisymmetric function, by means of a universal property. The terminology is taken from the case of posets. The basic facts and relationships between these objects hold in general.

Instances of the zeta polynomial are:

  1. The zeta polynomial of posets
  2. The order polynomial of posets
  3. The enriched polynomial of posets
  4. The chromatic polynomial of graphs
  5. The Tutte polynomial of graphs
  6. The Martin polynomial of eulerian graphs.
We will prove a general reciprocity theorem for zeta polynomials, based on simple Hopf algebra theory.

Instances of the flag quasisymmetric function are:

  1. The ab-index of graded posets
  2. The enumerator of posets partitions
  3. The enumerator of enriched poset partitions
  4. The chromatic symmetric function of a graph
  5. The q-chromatic symmetric function of Stanley
  6. A flag version of the Martin polynomial.

Speaker's Contact Info: aguiar(at-sign)crm.umontreal.ca

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Combinatorics Seminar, Mathematics Department, MIT, sara(at-sign)math.mit.edu

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