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Séminaire Lotharingien de Combinatoire, 78B.74 (2017), 12 pp.

# Brittney Ellzey

# Chromatic Quasisymmetric Functions of Directed Graphs

**Abstract.**
Chromatic quasisymmetric functions of labeled graphs were defined by
Shareshian and Wachs as a refinement of Stanley's chromatic symmetric
functions. In this extended abstract, we consider an extension of
their definition from labeled graphs to directed graphs, suggested by
Richard Stanley. We obtain an *F*-basis expansion of the chromatic
quasisymmetric functions of all digraphs and a *p*-basis expansion for
all *symmetric* chromatic quasisymmetric functions of digraphs,
extending work of Shareshian-Wachs and Athanasiadis. We show that the
chromatic quasisymmetric functions of proper circular arc digraphs are
symmetric functions, which generalizes a result of Shareshian and
Wachs on natural unit interval graphs. The directed cycle on n
vertices is contained in the class of proper circular arc digraphs,
and we give a generating function for the *e*-basis expansion of the
chromatic quasisymmetric function of the directed cycle, refining a
result of Stanley for the undirected cycle. We present a
generalization of the Shareshian-Wachs refinement of the
Stanley-Stembridge *e*-positivity conjecture.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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