  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.
