Related papers: Chromatic Quasisymmetric Class Functions for combi…
We generalize the notion of a coloring complex of a graph to linearized combinatorial Hopf monoids. We determine when a linearized combinatorial Hopf monoid has such a construction, and discover some inequalities that are satisfied by the…
We define vertex-colourings for edge-partitioned digraphs, which unify the theory of P-partitions and proper vertex-colourings of graphs. We use our vertex-colourings to define generalized chromatic functions, which merge the chromatic…
We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the $H$-coloring of a graph $G$ to define a generalization of Stanley's chromatic symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically equivalent…
We investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes.…
We introduce megagreedoids, which generalize polymatroids, megamatroids, and greedoids. We define a quasisymmetric function invariant for a megagreedoid, and show that it has a positive expansion in the basis of fundamental quasisymmetric…
The colored quasisymmetric functions, like the classic quasisymmetric functions, are known to form a Hopf algebra with a natural peak subalgebra. We show how these algebras arise as the image of the algebra of colored posets. To effect this…
This paper realizes of two families of combinatorial symmetric functions via the complex character theory of the finite general linear group $\mathrm{GL}_{n}(\mathbb{F}_{q})$: chromatic quasisymmetric functions and vertical strip LLT…
We provide a construction for the kromatic symmetric function $\overline{X}_G$ of a graph introduced by Crew, Pechenik, and Spirkl using combinatorial (linearly compact) Hopf algebras. As an application, we show that $\overline{X}_G$ has a…
The chromatic symmetric function $X_H$ of a hypergraph $H$ is the generating function for all colorings of $H$ so that no edge is monochromatic. When $H$ is an ordinary graph, it is known that $X_H$ is positive in the fundamental…
In this paper, we introduce and study two variants of the chromatic quasisymmetric function of a graph: the total chromatic quasisymmetric function via vertex labeling and via acyclic orientations. The original definition of the chromatic…
For a natural unit interval order $P$, we describe proper colorings of the incomparability graph of $P$ in the language of heaps. We also introduce a combinatorial operation, called a \emph{local flip}, on the heaps. This operation defines…
We study the chromatic symmetric function on graphs, and show that its kernel is spanned by the modular relations. We generalize this result to the chromatic quasisymmetric function on hypergraphic polytopes, a family of generalized…
In this article we show how to compute the chromatic quasisymmetric function of indifference graphs from the modular law introduced by Guay-Paquet. We provide an algorithm which works for any function that satisfies this law, such as…
We study the symmetric function and polynomial combinatorial invariants of Hopf algebras of permutations, posets and graphs. We investigate their properties and the relations among them. In particular, we show that the chromatic symmetric…
The Stanley chromatic symmetric function $X_G$ of a graph $G$ is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology…
We introduce a coloured generalization $\mathrm{NSym}_A$ of the Hopf algebra of non-commutative symmetric functions described as a subalgebra of the of rooted ordered coloured trees Hopf algebra. Its natural basis can be identified with the…
For a generalized permutohedron $Q$ the enumerator $F(Q)$ of positive lattice points in interiors of maximal cones of the normal fan $\Sigma_Q$ is a quasisymmetric function. We describe this function for the class of nestohedra as a Hopf…
For a hypergraphic polytope there is a weighted quasisymmetric function which enumerates positive integer points in its normal fan and determines its f-polynomial. This quasisymmetric function invariant of hypergraphs extends the Stanley…
Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize the chromatic polynomial and are related to Hessenberg varieties and diagonal harmonics. Motivated by the Stanley--Stembridge…
We introduce a superspace analogue of combinatorial Hopf algebras (Aguiar-Bergeron-Sottile, 2006), and show that the Hopf superalgebra of quasi-symmetric (resp. symmetric) functions in superspace (Fishel-Lapointe-Pinto, 2019) is a terminal…