When is the Chromatic Quasisymmetric Function Symmetric?

We investigate the problem of when a chromatic quasisymmetric function (CQF) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>G</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>;</mml:mo> <mml:mi>q</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of a graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> is in fact symmetric. We first prove the remarkable fact that if a product of two quasisymmetric functions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> in countably infinitely many variables is symmetric, then in fact <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> must be symmetric. This allows the problem to be reduced to the case of connected graphs. We then show that any labeled graph having more than one source or sink has a nonsymmetric CQF. As a corollary, we find that all trees other than a directed path have a nonsymmetric CQF. We also show that a family of graphs we call “mixed mountain graphs” always have symmetric CQF.