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A Chromatic Symmetric Function in Noncommuting Variables

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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).

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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