e-basis Coefficients of Chromatic Symmetric Functions

TL;DR

The paper extends Stanley's framework linking sums of $e$-basis coefficients of chromatic symmetric functions to acyclic orientations by introducing vertex-weighted and set-weighted graph models. It proves a vertex-weighted generalization of Stanley's Theorem 3.4 (Theorem main) using a refined sink-map and deletion-contraction approach, and it defines admissibility and maximality notions to connect coefficient sums $\sigma_{\mu,j}(X_{(G,\omega)})$ to signed counts over acyclic orientations and weight maps. A conjectured strengthening (Conjecture) broadens the combinatorial interpretation to a larger class of coefficient sums, notably proposing a weight-drop mechanism via generalized $2$-step weight maps with $s$-allowability, with supporting evidence in edgeless and two-vertex graphs. The work has potential implications for unweighted claw-free graphs, offering a path toward interpreting individual $e$-basis coefficients beyond current results such as Hikita's interpretation for unit interval graphs. Overall, the results advance the combinatorial understanding of $e$-positivity and coefficient interpretation in chromatic symmetric functions through vertex-weighted generalizations and conjectural broadening to claw-free graph classes.

Abstract

A well-known result of Stanley's shows that given a graph $G$ with chromatic symmetric function expanded into the basis of elementary symmetric functions as $X_G = \sum c_λe_λ$, the sum of the coefficients $c_λ$ for $λ$ with $λ_1' = k$ (equivalently those $λ$ with exactly $k$ parts) is equal to the number of acyclic orientations of $G$ with exactly $k$ sinks. However, more is known. The sink sequence of an acyclic orientation of $G$ is a tuple $(s_1,\dots,s_k)$ such that $s_1$ is the number of sinks of the orientation, and recursively each $s_i$ with $i > 1$ is the number of sinks remaining after deleting the sinks contributing to $s_1,\dots,s_{i-1}$. Equivalently, the sink sequence gives the number of vertices at each level of the poset induced by the acyclic orientation. A lesser-known follow-up result of Stanley's determines certain cases in which we can find a sum of $e$-basis coefficients that gives the number of acyclic orientations of $G$ with a given partial sink sequence. Of interest in its own right, this result also admits as a corollary a simple proof of the $e$-positivity of $X_G$ when the stability number of $G$ is $2$. In this paper, we prove a vertex-weighted generalization of this follow-up result, and conjecture a stronger version that admits a similar combinatorial interpretation for a much larger set of $e$-coefficient sums of chromatic symmetric functions. In particular, the conjectured formula would give a combinatorial interpretation for the sum of the coefficients $c_λ$ with prescribed values of $λ_1'$ and $λ_2'$ for any unweighted claw-free graph (not necessarily an incomparability graph, as in the setting of the Stanley-Stembridge conjecture).

Figures (17)

• Figure 1: $G$, including an isolated vertex
• Figure 2: Acyclic orientations of $G$, with first-level sinks colored in yellow
• Figure 3: A set-weighted graph $(G,\omega)$, where the set associated with each vertex is written next to the vertex label.
• Figure 4: Graph sequence formed by the 4-step weight sequence $S$
• Figure 5: Graph sequence formed by $S$ and $\gamma$

Theorems & Definitions (50)

• Theorem 1: Theorem 3.3, stanley
• Theorem 2: Theorem 3.4, stanley
• Definition 3: delconstanley
• Lemma 4: delcon, Lemma 2
• Definition 5
• Definition 6
• Lemma 7
• Definition 8
• Definition 9
• Definition 10