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 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. 1. Introduction The main ideas in this note are simple but require an inordinate number of definitions to state. In this section we skip most of these definitions so as not to obscure the exposition with a mass of technicalities. The missing definitions are given in the next section. Let G be a finite simple undirected graph and let XG be its chromatic symmetric function. Expand XG in terms of e..