Chromatic quasisymmetric functions and noncommutative $P$-symmetric functions

TL;DR

This paper explores the combinatorial structure of chromatic quasisymmetric functions related to unit interval orders, introducing new operations and establishing dualities that lead to positive expansions in symmetric function bases.

Contribution

It introduces a local flip operation on heaps, defines a noncommutative symmetric function analogue, and establishes a duality with chromatic quasisymmetric functions for unit interval orders.

Findings

Defined a local flip operation refining the ascent statistic

Established a duality between chromatic quasisymmetric functions and noncommutative symmetric functions

Provided partial results towards the e-positivity conjecture

Abstract

For a natural unit interval order $P$, we describe proper colorings of the incomparability graph of $P$ in the language of heaps. We also introduce a combinatorial operation, called a \emph{local flip}, on the heaps. This operation defines an equivalence relation on the proper colorings, and the equivalence relation refines the ascent statistic introduced by Shareshian and Wachs. In addition, we 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. In particular, we present some partial results for the $e$-positivity conjecture.