**Abstract:**

The chromatic symmetric function \(X_G\) of a graph \(G\) was introduced by Richard Stanley in 1995 as a generalization of the chromatic polynomial. The chromatic symmetric function has particularly interesting properties when \(G\) is the incomparability graph of a \((3+1)\)-free poset. In this case, \(X_G\) has positive integer coefficients when expanded in the basis of Schur functions, and the Stanley-Stembridge conjecture states that \(X_G\) has positive integer coefficients when expanded in the basis of elementary symmetric functions. For a \((3+1)\)-free poset \(P\), we define a ring of \(P\)-analogues of symmetric functions in noncommuting variables and reformulate the Stanley-Stembridge conjecture into a conjecture about this ring. With this formulation, we show that the coefficient of the elementary symmetric function \(e_\lambda\) is a non-negative integer when \(\lambda\) is a two-column, hook, or rectangular shape. This talk is based on joint work with Jonah Blasiak, Holden Eriksson, and Pavlo Pylyavskyy.

**Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.**

**Join Zoom Meeting: https://washington.zoom.us/j/ 91547335974Meeting ID: 915 4733 5974**