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In an effort to unify the theory of symmetric functions and the chromatic polynomial, Stanley introduced the chromatic symmetric function and studied its decomposition into different bases. The most notable result is that the chromatic symmetric function was shown to be Schur-positive for a family of graphs known as unit interval graphs. I will discuss a different combinatorial way of tackling this problem using crystals and talk about some open problems that this method could extend to.

This work is ongoing with Anne Schilling.