Relative symmetric polynomials and money change problem

This article is devoted to the number of non-negative solutions of the linear Diophantine equation
$$a_1t_1+a_2t_2+\cdots +a_nt_n=d,$$
where $a_1, \ldots, a_n$, and $d$ are positive integers. We obtain a relation between the number of solutions of this equation and characters of the symmetric group, using 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.