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

Chromatic quasisymmetric functions and noncommutative $P$-symmetric functions