Chromatic Quasisymmetric Class Functions of Linearized Combinatorial Hopf Monoids
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
White, Jacob A. 2022. “Chromatic Quasisymmetric Class Functions of Linearized Combinatorial Hopf Monoids.” In Proceedings of the 34th Conference on Formal Power Series and Algebraic Combinatorics, 12. Bangalore. https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2022/58.pdf.
Proceedings of the 34th Conference on Formal Power Series and Algebraic Combinatorics