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New interpretations for Kromatic symmetric function expansions

  • Aarush Kulkarni
  • Laura Pierson
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Abstract

The Kromatic symmetric function (KSF) X̅G, introduced by Crew, Pechenik, and Spirkl (2026), is a K-theoretic analogue of the chromatic symmetric function (CSF) XG. We study expansion formulas for the KSF in two different bases. First, we explore recursive ways to compute the KSF's expansion in the K-theoretic monomial symmetric function basis m̃̅λ, generalizing formulas that were used by the second author and Samanta to show that the KSF distinguishes certain families of graphs that are not distinguished by the CSF. Second, we give new interpretations for the KSF's expansion in the K-theoretic power sum basis p̅λ, using an inclusion-exclusion approach like the one from Stanley (1995) instead of an acyclic orientation approach. Finally, we give K-analogues of Schmitt's and of Humpert and Martin's antipode formulas for the Hopf algebra of graphs, along with a Hopf algebra interpretation for both the p̅-expansion and m̃̅-expansion of the KSF.

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Cite

BibTeXRIS
BibTeX
@misc{kulkarni2026interpretations,
  title = {{New interpretations for Kromatic symmetric function expansions}},
  author = {Aarush Kulkarni and Laura Pierson},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.38876},
  eprint = {2609.38876},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.38876},
}
RIS
TY  - UNPB
TI  - New interpretations for Kromatic symmetric function expansions
AU  - Aarush Kulkarni
AU  - Laura Pierson
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.38876
UR  - https://arxiv.org/abs/2609.38876
AB  - The Kromatic symmetric function (KSF) X̅_G, introduced by Crew, Pechenik, and Spirkl (2026), is a K-theoretic analogue of the chromatic symmetric function (CSF) X_G. We study expansion formulas for the KSF in two different bases. First, we explore recursive ways to compute the KSF's expansion in the K-theoretic monomial symmetric function basis m̃̅_λ, generalizing formulas that were used by the second author and Samanta to show that the KSF distinguishes certain families of graphs that are not distinguished by the CSF. Second, we give new interpretations for the KSF's expansion in the K-theoretic power sum basis p̅_λ, using an inclusion-exclusion approach like the one from Stanley (1995) instead of an acyclic orientation approach. Finally, we give K-analogues of Schmitt's and of Humpert and Martin's antipode formulas for the Hopf algebra of graphs, along with a Hopf algebra interpretation for both the p̅-expansion and m̃̅-expansion of the KSF.
ER  -