  Stanley has studied a symmetric function generalization X_G of the chromatic
polynomial of a graph G. The innocent-looking Stanley-Stembridge Poset Chain
Conjecture states that the expansion of X_G in terms of elementary symmetric
functions has nonnegative coefficients if G is a clawfree incomparability
graph. Here we give a combinatorial interpretation of these coefficients by
combining Gasharov's work on the conjecture with Egecioglu and Remmel's
combinatorial interpretation of the inverse Kostka matrix. This gives a new
proof of a partial nonnegativity result of Stanley. As an interesting byproduct
we derive a previously unnoticed result relating acyclic orientations to
P-tableaux.
