## Abstract

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 \({\mathcal{X}_G \left( n \right)}\) under a certain specialization of variables. He then proved various theorems generalizing results about \({\mathcal{X}_G \left( n \right)}\), 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).

## References

- 1.
A. Blass and B. Sagan, “Bijective proofs of two broken circuit theorems,”

*J. Graph Theory***10**(1986), 15–21. - 2.
T. Chow, personal communication.

- 3.
P. Doubilet, “On the foundations of combinatorial theory. VII: symmetric functions through the theory of distribution and occupancy,”

*Studies in Applied Math.***51**(1972), 377–396. - 4.
V. Gasharov, “Incomparability graphs of (

**3**+**1**)-free posets are*s*-positive,”*Discrete Math.***157**(1996), 193–197. - 5.
D. Gebhard, “Noncommutative symmetric functions and the chromatic polynomial,”

*Conference Proceedings at the 7th International Conference on Formal Power Series and Algebraic Combinatorics*, Université de Marne-la-Vallée, 1995. - 6.
D. Gebhard and B. Sagan, “Sinks in acyclic orientations of graphs,”

*J. Combin. Theory*(B)**80**(2000), 130–146. - 7.
I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh, and J.-I. Thibon, “Noncommutative symmetric functions,”

*Advances in Math.***112**(1995), 218–348. - 8.
C. Greene and T. Zaslavsky, “On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non-radon partitions, and orientations of graphs,”

*Trans. Amer. Math. Soc.***280**(1983), 97–126. - 9.
F. Harary and E. Palmer, “The reconstruction of a tree from its maximal subtrees,”

*Canad. J. Math.***18**(1966), 803–810. - 10.
R.P. Stanley, “Acyclic orientations of graphs,”

*Discrete Math.***5**(1973), 171–178. - 11.
R.P. Stanley, “A symmetric function generalization of the chromatic polynomial of a graph,”

*Advances in Math.***111**(1995), 166–194. - 12.
R.P. Stanley, “Graph colorings and related symmetric functions: Ideas and applications: A description of results, interesting applications, & notable open problems,” Selected papers in honor of Adriano Jarsia (Taormina, 1994)

*Discrete Math.***193**(1998), 267–286. - 13.
R.P. Stanley and J. Stembridge, “On immanants of Jacobi-Trudi matrices and permutations with restricted position,”

*J. Combin. Theory (A)***62**(1993), 261–279. - 14.
S.D. Noble and D.J.A.Welsh, “A weighted graph polynomial from chromatic invariants of knots,” Symposium àla Mémoire de François Jaeger (Grenoble, 1998)

*Annales l'Institut Fourier***49**(1999), 1057–1087.

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Gebhard, D.D., Sagan, B.E. A Chromatic Symmetric Function in Noncommuting Variables.
*Journal of Algebraic Combinatorics* **13, **227–255 (2001) doi:10.1023/A:1011258714032

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- chromatic polynomial
- deletion-contraction
- graph
- symmetric function in noncommuting variables