Séminaire Lotharingien de Combinatoire, 86B.58 (2022), 12 pp.
Jacob A. White
Chromatic Quasisymmetric Class Functions of Linearized Combinatorial Hopf Monoids
Abstract.
We study the chromatic quasisymmetric class function of a linearized combinatorial Hopf monoid. Given a linearized combinatorial Hopf monoid H, and an H-structure h on a set N, there are proper colorings of h, generalizing graph colorings and poset partitions. We show that the automorphism group of h acts on the set of proper colorings, which gives rise to the chromatic quasisymmetric class function. For the Hopf monoid of graphs this invariant generalizes Stanley's chromatic symmetric function and the orbital chromatic polynomial studied by Cameron and Kayibi.
We show that, under certain conditions, the chromatic quasisymmetric class function of h is the flag quasisymmetric class function of the coloring complex of h. We use this result to deduce various positivity results, and inequalities for the associated orbital polynomial invariants.
Received: November 25, 2021.
Accepted: March 4, 2022.
Final version: April 1, 2022.
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