Chromatic quasisymmetric functions and noncommutative $P$-symmetric functions

Byung-Hak Hwang

Published 2022-08-21Version 1

For a natural unit interval order $P$, we describe proper colorings of the incomparability graph of $P$ in the language of heaps. After then, we introduce a combinatorial operation, called a local flip, on the heaps. This operation defines an equivalence relation on the proper colorings, and the equivalent relation refines the ascent statistic introduced by Shareshian and Wachs. We also define an analogue of noncommutative symmetric functions introduced by Fomin and Greene, with respect to $P$. We establish a duality between the chromatic quasisymmetric function of $P$ and these noncommutative symmetric functions. This duality leads us to positive expansions of the chromatic quasisymmetric functions into several symmetric function bases. Also, we present some partial results for the $e$-positivity conjecture.