  We give a new (inductive) proof of the classical Frobenius--Young
correspondence between irreducible complex representations of the symmetric
group and Young diagrams, using the new approach, suggested in \cite{OV, VO},
to determining this correspondence. We also give linear relations between
Kostka numbers that follow from the decomposition of the restrictions of
induced representations to the previous symmetric subgroup. We consider a
realization of representations induced from Young subgroups in polylinear forms
and describe its relation to Specht modules.
