Three Simple Reduction Formulas for the Denumerant Functions

Let $A$ be a nonempty set of positive integers. The restricted partition function $p_A(n)$ denotes the number of partitions of $n$ with parts in $A$. When the elements in $A$ are pairwise relatively prime positive integers, Ehrhart, Sert\"oz-\"Ozl\"uk, and Brown-Chou-Shiue derived three reduction formulas for $p_A(n)$ for $A$ with three parameters. We extend their findings for general $A$ using the Bernoulli-Barnes polynomials.