A Two Parameter Chromatic Symmetric Function

We introduce and develop a two-parameter chromatic symmetric function for a simple graph G over the field of rational functions in q and t , Q (q, t). We derive its expansion in terms of the monomial symmetric functions, mλ, and present various correlation properties which exist between the two-parameter chromatic symmetric function and its corresponding graph. Additionally, for the complete graph G of order n, its corresponding two parameter chromatic symmetric function is the Macdonald polynomial Q(n). Using this, we develop graphical analogues for the expansion formulas of the two-row Macdonald polynomials and the two-row Jack symmetric functions. Finally, we introduce the “complement” of this new function and explore some of its properties.

• Type: Journal
• Year: 2007
• Where: COMBINATORICS
• Authors: Ellison-Anne Williams
• Keywords: Chromatic Symmetric Function, COMBINATORICS 2007, Monomial Symmetric Functions, Symmetric Functions