Some properties of the Redei-Berge function and related combinatorial Hopf algebras

TL;DR

This paper explores properties of the Redei-Berge symmetric function, extending its framework to new combinatorial Hopf algebras of posets and permutations, revealing deletion-contraction properties and digraph invariants.

Contribution

It introduces two new combinatorial Hopf algebras and defines Redei-Berge functions for them, expanding understanding of their properties and applications.

Findings

Redei-Berge function exhibits deletion-contraction properties similar to chromatic symmetric functions

Identifies digraph invariants detectable by the Redei-Berge function

Extends the framework of Redei-Berge functions to posets and permutations

Abstract

Stanley and Grinberg introduced the symmetric function associated to digraphs, called the Redei-Berge symmetric function. In [8] is shown that this symmetric function arises from a suitable structure of combinatorial Hopf algebra on digraphs. In this paper, we introduce two new combinatorial Hopf algebras of posets and permutations and define corresponding Redei-Berge functions for them. By using both theories, of symmetric functions and of combinatorial Hopf algebras, we prove many properties of the Redei-Berge function. These include some forms of deletion-contraction property, which make it similar to the chromatic symmetric function. We also find some invariants of digraphs that are detected by the Redei-Berge function.