A Note on a Combinatorial Interpretation of the e-Coefficients of the Chromatic Symmetric Function

Stanley has studied a symmetric function generalization XG of the chromatic polynomial of a graph G. The innocent-looking Stanley-Stembridge Poset Chain Conjecture states that the expansion of XG 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 Eg̃eciog̃lu 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.