MacMahon's partition identity and the coin exchange problem (Q1040836)

If \(a_1,a_2,\dots, a_m\) are positive integers, let \(S= S(a_1,a_2,\dots, a_m)\) denote the set of all linear combinations of the \(a_i\) with nonnegative coefficients. Holroyd's theorem states that the partitions of the natural number \(n\) satisfying the following conditions are equi-numerous: {\parindent=8mm \begin{itemize}\item[(i)]every part, and every difference of parts belongs to \(S\); \item[(ii)]every part has multiplicity belonging to \(S\); \item[(iii)]every part is a multiple of some element of \(S\), provided that the \(a_i\) satisfy an additional condition. \end{itemize}} The author provides a bijective proof of the equality of the number of partitions of \(n\) that satisfy conditions (ii), (iii), respectively. She thereby simplifies a portion of Holroyd's proof.