A Complete Multipartite Basis for the Chromatic Symmetric Function

In the vector space of symmetric functions, elements basis elementary functions are (up to a factor) chromatic disjoint unions cliques. We consider their graph complements, $\{r_{\lambda}: \lambda \text{ an integer partition}\}$ defined as complete multipartite graphs. This was first introduced by Penaguiao [21]. provide combinatorial interpretation for coefficients change-of-basis formula between $r_{\lambda}$ and monomial we show that Tutte $G$ when expanded in $r$-basis enumerate certain intersections partitions $V(G)$ into stable sets.