Some Properties of the Redei–Berge Function and Related Combinatorial Hopf Algebras

Abstract

Stanley and Grinberg introduced a symmetric function associated with digraphs, called the Redei–Berge symmetric function. In Grujić and Stojadinović (The Redei–Berge Hopf algebra of digraphs. Periodica Mathematica Hungarica. arXiv:2402.07606v2) is shown that this symmetric function arises from a suitable structure of a 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 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.