Related papers: A Noncommutative Chromatic Symmetric Function

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…

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…

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…

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…

Stanley has studied a symmetric function generalization X_G of the chromatic polynomial of a graph G. The innocent-looking Stanley-Stembridge Poset Chain Conjecture states that the expansion of X_G in terms of elementary symmetric functions…

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…

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…

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…

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…

In 1995, Richard Stanley introduced the chromatic symmetric function $X_G$ of a graph $G$ and proved that, when written in terms of the elementary symmetric functions, it reveals the number of acyclic orientations of $G$ with a given number…

We give a new proof of Chung and Graham's ``G-descent expansion'' of the classical chromatic polynomial, as well as a special case of the quasi-symmetric function expansion of the path-cycle symmetric function Xi_D. Both proofs rely on…

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.…

If we consider previously introduced extensions of Stanley's chromatic symmetric function $X_{G}(x_1, x_2, \ldots)$ for a graph $G$ to elements in the algebra $\textsf{QSym}$ of quasisymmetric functions and in the algebra $\textsf{NCSym}$…

We extend the definition of the chromatic symmetric function $X_G$ to include graphs $G$ with a vertex-weight function $w : V(G) \rightarrow \mathbb{N}$. We show how this provides the chromatic symmetric function with a natural…

Stanley's symmetrized chromatic polynomial is a generalization of the ordinary chromatic polynomial to a graph invariant with values in a ring of polynomials in infnitely many variables. The ordinary chromatic polynomial is a specialization…

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…

We discover new linear relations between the chromatic symmetric functions of certain sequences of graphs and apply these relations to find new families of e-positive unit interval graphs. Motivated by the results of Gebhard and Sagan, we…

Let $\mathfrak g$ be a Borcherds algebra with the associated graph $G$. We prove that the chromatic symmetric function of $G$ can be recovered from the Weyl denominator identity of $\mathfrak g$ and this gives a Lie theoretic proof of…

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated…