  This article is devoted to the number of non-negative solutions of the linear
Diophantine equation $$ a_1t_1+a_2t_2+... a_nt_n=d, $$ where $a_1, ..., a_n$,
and $d$ are positive integers. We obtain a relation between the number of
solutions of this equation and characters of the symmetric group, using {\em
relative symmetric polynomials}. As an application, we give a necessary and
sufficient condition for the space of the relative symmetric polynomials to be
non-zero.
