  In a series of recent talks Richard Stanley introduced a symmetric function
associated to digraphs called the Redei-Berge symmetric function. This
symmetric function enumerates descent sets of permutations corresponding to
digraphs. We show that such constructed symmetric function arises from a
suitable structure of combinatorial Hopf algebra on digraphs. The induced
Redei-Berge polynomial satisfies the deletion-contraction property which makes
it similar to the chromatic polynomial. The Berge's classical result on the
number of Hamiltonian paths in digraphs is a consequence of the reciprocity
formula for the Redei-Berge polynomial.
