Chromatic bases for symmetric functions

Richard Stanley introduced the chromatic symmetric function \$X_G\$ of a simple graph \$G\$, which is the sum of all possible proper colorings with colors \$\{1,2,3,\ldots \}\$ coded as monomials in commuting variables. These formal power series are symmetric functions and generalize the chromatic polynomial. Soojin Cho and Stephanie van Willigenburg found that, given a sequence of connected graphs \$G_1,G_2, \ldots\$ where \$G_i\$ has \$i\$ vertices, \$\{X_{G_i}\}\$ forms a basis for the algebra of symmetric functions. This provides a multitude of new bases since they also discovered that only the sequence of complete graphs provides a basis that is equivalent to a classical basis, namely the elementary symmetric functions. This talk will discuss new results on chromatic symmetric functions using these new and old bases, and additionally we will also resolve Stanley's \$e\$-Positivity of Claw-Contractible-Free Graphs. This is joint work with Angele Hamel and Stephanie van Willigenburg.