Geodesic
A chromatic symmetric function in noncommuting variables
Journal of Algebraic Combinatorics, Tome 13 (2001) no. 3, pp. 227-255.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $X _{ G} ( n )$ mathcalX_G $left( n right)$} under a certain specialization of variables. He then proved various theorems generalizing results about $X _ G ( n )$ {\mathcal{X}\_G $left( n \right)$ , 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).
Keywords: chromatic polynomial, deletion-contraction, graph, symmetric function in noncommuting variables 
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     title = {A chromatic symmetric function in noncommuting variables},
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Gebhard, David D.; Sagan, Bruce E. A chromatic symmetric function in noncommuting variables. Journal of Algebraic Combinatorics, Tome 13 (2001) no. 3, pp. 227-255. http://geodesic.mathdoc.fr/item/JAC_2001__13_3_a4/