Combinatorial expressions of Hopf polynomial invariants

TL;DR

This paper extends the combinatorial interpretation of Hopf polynomial invariants to both positive and negative integers across various combinatorial structures, providing new formulas and proofs.

Contribution

It offers a unified combinatorial interpretation of Hopf polynomial invariants over positive and negative integers for multiple combinatorial Hopf monoids, including hypergraphs and permutahedra.

Findings

Provides a combinatorial interpretation over negative integers for hypergraphs and permutahedra.

Introduces two different proofs for the negative integer interpretation in hypergraphs.

Extends results to graphs, simplicial complexes, and building sets.

Abstract

In 2017 Aguiar and Ardila provided a generic way to construct polynomial invariants of combinatorial objects using the notions of Hopf monoids and characters of Hopf monoids. They show that it is possible to find a combinatorial interpretation of these polynomials over negative integers using the antipode and give a cancellation-free grouping-free formula for the antipode on generalized permutahedra. In this work, we give a combinatorial interpretation of these polynomials over both positive integers and negative integers for the Hopf monoids of generalized permutahedra and hypergraphs and for every character on these two Hopf monoids. In the case of hypergraphs, we present two different proofs for the interpretation on negative integers. We then deduce similar results on other combinatorial objects including graphs, simplicial complexes and building sets.