A Chromatic Symmetric Function in Noncommuting Variables

Stanley (Advances in Math. 111, 1995, 166–194) associated with a graph G a symmetric function X _G which reduces to G's chromatic polynomial under a certain specialization of variables. He then proved various theorems generalizing results about , as well as new ones that cannot be interpreted on the level of the chromatic polynomial. Unfortunately, X _G does not satisfy a Deletion-Contraction Law which makes it difficult to apply the useful technique of induction. We introduce a symmetric function Y _G in noncommuting variables which does have such a law and specializes to X _G when the variables are allowed to commute. This permits us to further generalize some of Stanley's theorems and prove them in a uniform and straightforward manner. Furthermore, we make some progress on the (3 + 1)-free Conjecture of Stanley and Stembridge (J. Combin Theory (A) J. 62, 1993, 261–279).