Power sum expansions for Kromatic symmetric functions using Lyndon heaps
Laura Pierson
TL;DR
The paper develops a comprehensive $K$-theoretic lift of the chromatic symmetric-function framework by introducing the Kromatic symmetric function $\overline{X}_G$ and two $K$-basis variants $\overline{p}_\lambda$ and $\overline{p}'_\lambda$. Using Lyndon heaps as canonical representatives of acyclic-orientation classes on clan graphs, the authors give combinatorial interpretations for the $\overline{p}$-coefficients of $\overline{X}_G$ and its $\omega$-image, and they prove cleaner $\overline{p}'$-expansions. A key structural result is that $\overline{X}_G$ encodes the multiset of independence polynomials of all induced subgraphs of $G$, enabling concise proofs that such data determine induced-subgraph counts; the work also extends to quasisymmetric analogues $\overline{X}_G(q)$ for unit interval order incomparability graphs. The paper provides bijective proofs of heap-series factorizations and derives transitions between $p$-bases to obtain quasisymmetric power-sum expansions, including both $p$- and $p'$-type formulas and their $q$-deformations, with specializations to $q=1$ recovering the classical results. These developments yield a rich combinatorial toolkit for analyzing subgraph structure and symmetry bases in the K-analogous setting.
Abstract
In arXiv:2301.02177, Crew, Pechenik, and Spirkl defined the Kromatic symmetric function $\overline{X}_G$ as a $K$-analogue of Stanley's chromatic symmetric function $X_G$, and one question they asked was how $\overline{X}_G$ expands in their $\overline{p}_λ$ basis, which they defined as a $K$-analogue of the classic power sum basis $p_λ.$ In arXiv:2408.01395, we gave a formula that partially answered this question but did not explain the combinatorial significance of the coefficients. Here, we give combinatorial descriptions for the $\overline{p}$-coefficients of $\overline{X}_G$ and $ω(\overline{X}_G)$, lifting the $p$-expansion of $X_G$ in terms of acyclic orientations that was given by Bernardi and Nadeau in arXiv:1904.01262. We also propose an alternative $K$-analogue $\overline{p}'$ of the $p$-basis that gives slightly cleaner expansion formulas. Our expansions are based on Lyndon heaps, introduced by Lalonde (1995), which are representatives for certain equivalence classes of acyclic orientations on clan graphs of $G$. Additionally, we show that knowing $\overline{X}_G$ is equivalent to knowing the multiset of independence polynomials of induced subgraphs of $G$, which gives shorter proofs of all our results from arXiv:2403.15929 that $\overline{X}_G$ can be used to determine the number of copies in $G$ of certain induced subgraphs. We also give power sum expansions for the Kromatic quasisymmetric function $\overline{X}_G(q)$ defined by Marberg in arXiv:2312.16474 in the case where $G$ is the incomparability graph of a unit interval order.