Abstract
To an extended generalized permutohedron we associate the weighted integer points enumerator, whose principal specialization is the f-polynomial. In the case of poset cones it refines Gessel?s P-partitions enumerator. We show that this enumerator is a quasisymmetric function obtained by universal morphism from the Hopf algebra of posets.
Highlights
In the seminal paper of Aguiar, Bergeron and Sottile [2] the notion of combinatorial Hopf algebra was introduced. They explained the ubiquity of quasisymmetric functions as generating functions in enumerative combinatorics
The integer points enumerator associated to a generalized permutohedron is a quasisymmetric function
We prove that the integer points enumerator associated to a poset cone coincides with the universal morphism from the Hopf algebra of posets to quasisymmetric functions
Summary
In the seminal paper of Aguiar, Bergeron and Sottile [2] the notion of combinatorial Hopf algebra was introduced. This result is analogous to the previous results for simple graphs [7], matroids and building sets [8], and spreads their validity to the case of extended generalized permutohedra. We provide an example of posets with the same P-partitions enumerators but which are distinguished by corresponding weighted quasisymmetric enumerators
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