  To a word $w$, we associate the rational function $\Psi_w = \prod (x_{w_i} -
x_{w_{i+1}})^{-1}$. The main object, introduced by C. Greene to generalize
identities linked to Murnaghan-Nakayama rule, is a sum of its images by certain
permutations of the variables. The sets of permutations that we consider are
the linear extensions of oriented graphs. We explain how to compute this
rational function, using the combinatorics of the graph $G$. We also establish
a link between an algebraic property of the rational function (the
factorization of the numerator) and a combinatorial property of the graph (the
existence of a disconnecting chain).
