A signed $e$-expansion of the chromatic quasisymmetric function

Foster Tom

A signed $e$-expansion of the chromatic quasisymmetric function

Combinatorics 2024-05-09 v2

Authors: Foster Tom

Abstract

We prove a new signed elementary symmetric function expansion of the chromatic quasisymmetric function of any natural unit interval graph. We then use a sign-reversing involution to prove a new combinatorial formula for K-chains, which are graphs formed by joining cliques at single vertices. This formula immediately implies $e$ -positivity and $e$ -unimodality for K-chains. We also prove a version of our signed $e$ -expansion for arbitrary graphs.

Keywords

symmetric functions graph polynomials graph theory

Cite

@article{arxiv.2311.08020,
  title  = {A signed $e$-expansion of the chromatic quasisymmetric function},
  author = {Foster Tom},
  journal= {arXiv preprint arXiv:2311.08020},
  year   = {2024}
}

Comments

36 pages, 13 figures. Replaced previous version to prove the q-analogue and simplify the proofs

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