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

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.