Relative symmetric polynomials and money change problem

This article is devoted to the number of non- negative solutions of the linear Diophantine equation a₁t₁ + a₂t₂ + · · · + a_nt_n = d, where a₁,..., 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.