Relative symmetric polynomials and money change problem

TL;DR

This paper explores the connection between the number of solutions to a linear Diophantine equation and characters of the symmetric group, utilizing relative symmetric polynomials to establish new relations and conditions.

Contribution

It introduces a novel relation between solutions of Diophantine equations and symmetric group characters using relative symmetric polynomials, and provides conditions for the non-vanishing of related polynomial spaces.

Findings

Derived a relation between solutions count and symmetric group characters

Established a necessary and sufficient condition for the non-zero space of relative symmetric polynomials

Connected Diophantine solutions to algebraic structures in symmetric polynomial theory

Abstract

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.