The Chromatic Quasisymmetric Class Function of a Digraph

White, Jacob

doi:10.1007/s00026-021-00556-1

Cited by 1 publication

(5 citation statements)

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“…We remark that the result for posets is already a special case for similar results we obtained previously for double posets [21]. Similarly, the result for graphs can also be deduced from previous results we have for digraphs [21].…”

Section: Main Theorem and Examplessupporting

confidence: 89%

“…As an example, if we let d be the double poset in Our Theorem 22 applies. In this case, several of the results were previously obtained in [21].…”

Section: Double Posetsmentioning

confidence: 57%

“…In Section 2, we discuss quasisymmetric class functions and polynomial class functions. This material was studied previously in [21]. We show that the property of being M -increasing for a quasisymmetric class function leads to various inequalities for the f -vector of the principal specialization.…”

Section: The Orbital Polynomial ψ Omentioning

confidence: 89%

“…An orbital version of the order polynomial of a poset was studied by Jochemko [15], and a quasisymmetric function generalization for double posets was studied by Grinberg [13]. We generalized further to a quasisymmetric class function associated to a double poset [21], where we also proved an analogue of Theorem 1. Hence, we are after a general framework for constructing such quasisymmetric functions, and proving theorems about them.…”

Section: Introductionmentioning

confidence: 85%

“…This quasisymmetric function is studied extensively by Grinberg [13], who shows F -positivity results when the double poset is tertispecial. We introduced the corresponding quasisymmetric class function in [21]. Now we show how the same function comes from linearized combinatorial Hopf monoids.…”

Section: Double Posetsmentioning

confidence: 99%

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Chromatic Quasisymmetric Class Functions for combinatorial Hopf monoids

White¹

2021

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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.

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Section: Main Theorem and Examplessupporting

confidence: 89%

“…As an example, if we let d be the double poset in Our Theorem 22 applies. In this case, several of the results were previously obtained in [21].…”

Section: Double Posetsmentioning

confidence: 57%

Section: The Orbital Polynomial ψ Omentioning

confidence: 89%