  It is an important problem in algebraic combinatorics to deduce the Schur
function expansion of a symmetric function whose expansion in terms of the
fundamental quasisymmetric function is known. For example, formulas are known
for the fundamental expansion of a Macdonald symmetric function and for the
plethysm of two Schur functions, while the Schur expansions of these
expressions are still elusive. Egge, Loehr and Warrington provided a method to
obtain the Schur expansion from the fundamental expansion by replacing each
quasisymmetric function by a Schur function (not necessarily indexed by a
partition) and using straightening rules to obtain the Schur expansion. Here we
provide a new method that only involves the coefficients of the quasisymmetric
functions indexed by partitions and the quasi-Kostka matrix. As an application,
we identify the lexicographically largest term in the Schur expansion of the
plethysm of two Schur functions. We provide the Schur expansion of
$s_w[s_h](x,y)$ for $w=2,3,4$ using novel symmetric chain decompositions of
Young's lattice for partitions in a $w\times h$ box. For $w=4$, this is first
known combinatorial expression for the coefficient of $s_{\lambda}$ in
$s_{w}[s_{h}]$ for two-row partitions $\lambda$, and for $w=3$ the
combinatorial expression is new.
