  The Chip-firing game is a discrete dynamical system played on a graph, in
which chips move along edges according to a simple local rule. Properties of
the underlying graph are of course useful to the understanding of the game, but
since a conjecture of Biggs that was proved by Merino L\'opez, we also know
that the study of the Chip-firing game can give insights on the graph. In
particular, a strong relation between the partial Tutte polynomial $T_G(1,y)$
and the set of recurrent configurations of a Chip-firing game (with a
distinguished sink vertex) has been established for undirected graphs. A direct
consequence is that the generating function of the set of recurrent
configurations is independent of the choice of the sink for the game, as it
characterizes the underlying graph itself. In this paper we prove that this
property also holds for Eulerian directed graphs (digraphs), a class on the way
from undirected graphs to general digraphs. It turns out from this property
that the generating function of the set of recurrent configurations of an
Eulerian digraph is a natural and convincing candidate for generalizing the
partial Tutte polynomial $T_G(1,y)$ to this class. Our work also gives some
promising directions of looking for a generalization of the Tutte polynomial to
general digraphs.
