$α$-chromatic symmetric functions

TL;DR

The paper introduces the α-chromatic symmetric functions $\chi^{(\alpha)}_\pi[X;q]$ by the plethystic substitution $\chi_\pi[Q_\alpha X;q]$ with $Q_\alpha=\dfrac{q^\alpha-1}{q-1}$, generalizing the Shareshian–Wachs framework and linking to unicellular LLT polynomials. It provides positive combinatorial formulas: a monomial expansion in the basis $\{\alpha+kn_q\}$ and a falling-factorial expansion in terms of chromatic functions, along with Schur-positivity and symmetry results, and a connection to $q$-rook theory that yields a new solution to the $q$-hit problem. The work further explores the relation to LLT polynomials, offers an XY-technique to manipulate plethystic expressions, and discusses geometric interpretations via Hessenberg varieties and projective-space products, suggesting broad implications for Schur coefficients and Jack-polynomial analogies. Overall, the results give new combinatorial and algebraic tools for $\chi^{(\alpha)}_\pi$ and deepen connections among chromatic symmetry, LLT theory, rook theory, and geometry. The combination of explicit expansions, positivity results, and geometric ties positions these functions as a versatile bridge across several areas in algebraic combinatorics.

Abstract

In this paper, we introduce the \emph{$α$-chromatic symmetric functions} $χ^{(α)}_π[X;q]$, extending Shareshian and Wachs' chromatic symmetric functions with an additional real parameter $α$. We present positive combinatorial formulas with explicit interpretations. Notably, we show an explicit monomial expansion in terms of the $α$-binomial basis and an expansion into certain chromatic symmetric functions in terms of the $α$-falling factorial basis. Among various connections with other subjects, we highlight a significant link to $q$-rook theory, including a new solution to the $q$-hit problem posed by Garsia and Remmel in their 1986 paper introducing $q$-rook polynomials.

Figures (5)

• Figure 1: A Dyck path $\pi$ and corresponding $P(\pi)$ and $G(\pi)$.
• Figure 2: On the left, a placement of $4$ nonattacking rooks (denoted by bullets) on $B(2,3,5,5,5,6)$ with $\operatorname{inv}=8$. On the right, a placement of $6$ rooks on $[n]\times [n]$ with $2$ rooks on $B(0,1,3,4,4,4)$, where $\operatorname{stat}_{B}=10$.
• Figure 3: On the left, a proper coloring of $G_{(3,3,4,6,6,6,8,9,9)}$, and on the right, the corresponding coloring of $G_{\beta (\pi,w)}$.
• Figure 4: On the right, the Ferrers board $B_{\pi}=B(c_1(\pi),\ldots ,c_n(\pi)) = B(0,0,1,1,3,3,5)$ for $\pi =(2,4,4,6,6,7,7)$, with the relations in the poset $P_{\pi}$ appearing above $\pi$ on the left
• Figure 5: A placement of rooks $L_{\sigma}$ on the left. Here $\sigma=52416837$, with $\text{inv}_{\pi}(\sigma ^{-1})=5$, corresponding to the blue dots on the right.

Theorems & Definitions (46)

• Example 2.1
• Proposition 2.2
• Definition 2.3
• Proposition 2.4
• Definition 2.5
• Proposition 2.6
• proof
• Remark 2.7
• Theorem 2.8
• Theorem 2.9