# Integer points enumerator of hypergraphic polytopes

## TL;DR

This paper introduces a quasisymmetric function invariant for hypergraphic polytopes that counts integer points and generalizes the Stanley chromatic symmetric function for graphs.

## Contribution

It establishes a new combinatorial Hopf algebra framework linking hypergraphs to quasisymmetric functions, extending known graph invariants.

## Key findings

- Defines a weighted quasisymmetric function for hypergraphic polytopes
- Shows the invariant extends the Stanley chromatic symmetric function
- Proves the universal morphism to quasisymmetric functions coincides with the enumerator

## Abstract

For a hypergraphic polytope there is a weighted quasisymmetric function which enumerates positive integer points in its normal fan and determines its f-polynomial. This quasisymmetric function invariant of hypergraphs extends the Stanley chromatic symmetric function of simple graphs. We consider a certain combinatorial Hopf algebra of hypergraphs and show that universal morphism to quasisymmetric functions coincides with this enumerator function.