Related papers: H-chromatic symmetric functions
Stanley [9] introduced the chromatic symmetric function ${\bf X}_G$ associated to a simple graph $G$ as a generalization of the chromatic polynomial of $G$. In this paper we present a novel technique to write ${\bf X}_G$ as a linear…
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…
The chromatic symmetric function $X_G$ of a graph $G$ was introduced by Stanley. In this paper we introduce a quasisymmetric generalization $X^k_G$ called the $k$-chromatic quasisymmetric function of $G$ and show that it is positive in the…
The chromatic symmetric function $X_G$ is a power series that encodes the proper colorings of a graph $G$ by assigning a variable to each color and a monomial to each coloring such that the power of a variable in a monomial is the number of…
Stanley associated with a graph G a symmetric function X_G which reduces to G's chromatic polynomial under a certain specialization of variables. He then proved various theorems generalizing results about the chromatic polynomial, as well…
In this paper, we extend the chromatic symmetric function $X$ to a chromatic $k$-multisymmetric function $X_k$, defined for graphs equipped with a partition of their vertex set into $k$ parts. We demonstrate that this new function retains…
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…
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…
Richard Stanley defined the chromatic symmetric function $X_G$ of a graph $G$ and asked whether there are non-isomorphic trees $T$ and $U$ with $X_T=X_U$. We study variants of the chromatic symmetric function for rooted graphs, where we…
In 1995, Stanley introduced the well-known chromatic symmetric function $X_{G}(x_{1},x_{2},\ldots)$ of a graph $G$. It is a sum of monomial symmetric functions such that for each vertex coloring of $G$ there is exactly one of these…
Stanley introduced the chromatic symmetric function of a simple graph, which is a generalization of a chromatic polynomial. This is expressed in terms of the integer points of the complements of the corresponding graphic arrangement.…
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…
In Chapter 2 we study the path-cycle symmetric function of a digraph, a symmetric function generalization of Chung and Graham's cover polynomial. Most of this material appears in either Advances in Math. 118 (1996), 71-98 or J. Algebraic…
The chromatic symmetric $X_G$ function is a symmetric function generalization of the chromatic polynomial of a graph, introduced by Stanley (1995). Stanley gave an expansion formula for $X_G$ in terms of the power sum symmetric functions…
The chromatic symmetric function of a graph is a generalization of the chromatic polynomial. The key motivation for studying the structure of a chromatic symmetric function is to answer positivity conjectures by Stanley in 1995 and Gasharov…
In 1995 Stanley introduced a generalization of the chromatic polynomial of a graph $G$, called the chromatic symmetric function, $X_G$, which was generalized to noncommuting variables, $Y_G$, by Gebhard-Sagan in 2001. Recently there has…
We study the $H$-chromatic symmetric functions $X_G^H$ (introduced in (arXiv:2011.06063) as a generalization of the chromatic symmetric function (CSF) $X_G$), which track homomorphisms from the graph $G$ to the graph $H$. We focus first on…
The chromatic symmetric function $X_G$ is a sum of monomials corresponding to proper vertex colorings of a graph $G$. Crew, Pechenik, and Spirkl (2023) recently introduced a $K$-theoretic analogue $\overline{X}_G$ called the Kromatic…
Schur functions are a basis of the symmetric function ring that represent Schubert cohomology classes for Grassmannians. Replacing the cohomology ring with $K$-theory yields a rich combinatorial theory of inhomogeneous deformations, where…
In this note we obtain numerous new bases for the algebra of symmetric functions whose generators are chromatic symmetric functions. More precisely, if $\{ G_ k \}_{k\geq 1}$ is a set of connected graphs such that $G_k$ has $k$ vertices for…