Discrete Applied Math Seminar by Samantha Dahlberg: Chromatic Symmetric Functions: An Introduction and the Property of E-positivity, Part 2

Speaker: Samantha Dahlberg, Illinois Institute of Technology

Title: Chromatic Symmetric Functions: An introduction and the property of e-positivity

Abstract:

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. 

The first part of the presentation will introduce chromatic symmetric functions, their basic properties and what questions are studied. It will be accessible to anyone with basic knowledge of graph theory. 

The second part will focus on the property of e-positivity, when X_G can be written as a non-negative sum of elementary symmetric functions. The property of e-positivity has received a lot of attention lately, and is connected to Stanley and Stembridge's (3+1)-free conjecture. We will discuss what is known about e-positivity, new families of graphs that have been shown to be e-positive, the e-positivity of trees, and the resolution of Stanley's e-Positivity of Claw-Contractible-Free Graphs. This is joint work with Angele Foley, Adrian She, and Stephanie van Willigenburg.

Discrete Math Seminar

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