A Note on a Combinatorial Interpretation of thee - Coe cients of the Chromatic Symmetric

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 coeecients if G is a clawfree incomparabil-ity graph. Here we present a new approach that combines Gasharov's work on the conjecture with E~ gecio~ glu and Remmel's combinatorial interpretation of the inverse Kostka matrix. An interesting byproduct of our arguments is a connection, apparently new, between acyclic orientations and P-tableaux.