  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.
