Power sum expansions for Kromatic symmetric functions using Lyndon heaps

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 $\overline{p}_\lambda$ basis, which they defined as a $K$-analogue of the classic power sum basis $p_\lambda.$ 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 $\omega(\overline{X}_G)$, 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.