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A Noncommutative Chromatic Symmetric Function

David D. Gebhard·Bruce E. Sagan
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Abstract

Stanley 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 the chromatic polynomial, as well as new ones that cannot be interpreted at that level. Unfortunately, X_G does not satisfy a Deletion-Contraction Law which makes it difficult to apply induction. We introduce a symmetric function 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.

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David D. Gebhard, Bruce E. Sagan. 1999-07-12. A Noncommutative Chromatic Symmetric Function. https://arxiv.org/abs/math/9907076

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