# Weighted $\mathsf{P}-$partitions enumerator

## TL;DR

This paper introduces a weighted integer points enumerator for extended permutohedra, connecting it to quasisymmetric functions and refining existing enumerators for poset cones, with implications for combinatorial enumeration.

## Contribution

It defines a new weighted enumerator linked to extended permutohedra and shows it as a universal morphism from the Hopf algebra of posets, refining Gessel's P-partitions enumerator.

## Key findings

- The enumerator generalizes Gessel's P-partitions enumerator.
- It is a quasisymmetric function derived from a universal morphism.
- The principal specialization yields the f-polynomial of the extended permutohedron.

## Abstract

To an extended 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 $\mathsf{P}$-partitions enumerator. We show that this enumerator is a quasisymmetric function obtained by universal morphism from the Hopf algebra of posets.