Mathematical and Statistical Sciences Faculty Publications and Presentations

Document Type

Article

Publication Date

12-17-2021

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. The chromatic quasisymmetric class function enumerates the fixed points of this action, weighting each coloring with a monomial. For the Hopf monoid of graphs this invariant generalizes Stanley's chromatic symmetric function and specializes to the orbital chromatic polynomial of Cameron and Kayibi.

We also introduce the flag quasisymmetric class function of a balanced relative simplicial complex equipped with a group action. We show that, under certain conditions, the chromatic quasisymmetric class function of h is the flag quasisymmetric class function of a balanced relative simplicial complex that we call the coloring complex of h. We use this result to deduce various inequalities for the associated orbital polynomial invariants. We apply these results to several examples related to enumerating graph colorings, poset partitions, generic functions on matroids or generalized permutohedra, and others.

Comments

Preprint submitted to Elsevier

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Included in

Mathematics Commons

Share

COinS
 
 

To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.