  We describe a combinatorial formula for the coefficients when the dual
immaculate quasisymmetric functions are decomposed into Young quasisymmetric
Schur functions. We prove this using an analogue of Schensted insertion. Using
this result, we give necessary and sufficient conditions for a dual immaculate
quasisymmetric function to be symmetric. Moreover, we show that the product of
a Schur function and a dual immaculate quasisymmetric function expands
positively in the Young quasisymmetric Schur basis. We also discuss the
decomposition of the Young noncommutative Schur functions into the immaculate
functions. Finally, we provide a Remmel-Whitney-style rule to generate the
coefficients of the decomposition of the dual immaculates into the Young
quasisymmetric Schurs algorithmically and an analogous rule for the
decomposition of the dual bases.
