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arXiv · 2302.05992

An Extension of Stanley's Symmetric Acyclicity Theorem to Signed Graphs

Oscar Coppola·Jake Huryn·Michael Reilly
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Abstract

In 1995, Richard Stanley introduced the chromatic symmetric function $X_G$ of a graph $G$ and proved that, when written in terms of the elementary symmetric functions, it reveals the number of acyclic orientations of $G$ with a given number of sinks. In this paper, we generalize this result to signed graphs, that is, to graphs whose edges are labeled with $+$ or $-$ and whose colorings and orientations can interact with their signs. Additionally, we introduce a non-homogeneous basis which detects the number of sinks and which not only gives a Stanley-type result for signed graphs but gives an analogous result of this form for unsigned graphs as well.

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Oscar Coppola, Jake Huryn, Michael Reilly. 2023-02-12. An Extension of Stanley's Symmetric Acyclicity Theorem to Signed Graphs. https://arxiv.org/abs/2302.05992

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