A Two Parameter Chromatic Symmetric Function

Ellison-Anne Williams

Abstract


We introduce and develop a two-parameter chromatic symmetric function for a simple graph $G$ over the field of rational functions in $q$ and $t,\,{\Bbb Q}(q,t)$. We derive its expansion in terms of the monomial symmetric functions, $m_{\lambda}$, 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.


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