  Skewing operators play a central role in the symmetric function theory
because of the importance of the product structure of the symmetric function
space. The theory of noncommutative symmetric functions is a useful tool for
studying expansions of a given symmetric function in terms of various bases. In
this paper, we establish a further development of the theory for studying
skewing operators. Using this machinery, we are able to easily reproduce the
Littlewood--Richardson rule, and provide recurrence relations for chromatic
quasisymmetric functions, which generalizes Harada--Precup's recurrence.
