Chromatic symmetric functions and e-positivity

Richard Stanley introduced the chromatic symmetric function X_G of a simple graph G, an algebraic encoding of all possible proper colorings with colors {1,2,3, ... }. These formal power series are symmetric functions that generalize the chromatic polynomial. In this talk we discuss the algebraic property of e-positivity, when X_G can  be written as a non-negative sum of elementary symmetric functions. We will also discuss what is known about e-positivity, resolve Stanley's e-Positivity of Claw-Contractible-Free Graphs (joint work with Angele Foley and Stephanie van Willigenburg), and make progress on Stanley and Stembridge’s (3+1)-free conjecture.