Chromatic quasisymmetric functions of the path graph

TL;DR

The paper resolves when the chromatic quasisymmetric function $X(P_n; \mathbf{x}, q)$ of a labeled path is symmetric, proving symmetry holds if and only if the labeling is the natural labeling (or its reverse). The authors develop a ribbon-tableaux framework via $RD(P_n)$ to translate colorings into ascent patterns and subribbon configurations, enabling a precise obstruction analysis for non-natural labelings. They show that non-natural labelings force unequal coefficients in the monomial basis $M_\alpha$, hence non-symmetry, and they extend the discussion to other trees, including the star graph, where symmetry fails in general; they also provide broader criteria for non-palindromicity in bipartite graphs. Overall, the work clarifies the symmetry landscape of CQFs for trees and highlights the role of ribbon combinatorics in understanding these symmetric properties.

Abstract

We show that the chromatic quasisymmetric function (CQF) of a labeled path graph on $n$ vertices is not symmetric unless the labeling is the natural labeling $1, 2, ..., n$ or its reverse $n, ..., 2, 1$. We also show that the star graph $K_{1, n-1}$ with $n\geq 3$ has a nonsymmetric CQF for all labelings.

Theorems & Definitions (32)

• Theorem 1.1
• Definition 2.1
• Proposition 2.2
• proof
• Corollary 2.3
• proof
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Proposition 4.1