Partition Identities and the Coin Exchange Problem

TL;DR

This paper explores partition identities related to the coin exchange problem, generalizing classical results and extending them to multiple integers under specific conditions.

Contribution

It introduces new partition identities connecting divisibility conditions with expressibility as combinations of integers a and b, generalizing MacMahon and Andrews' identities.

Findings

Partition identities for two integers a and b.

Generalization to multiple integers under certain conditions.

Connections between divisibility and expressibility in partitions.

Abstract

The number of partitions of n into parts divisible by a or b equals the number of partitions of n in which each part and each difference of two parts is expressible as a non-negative integer combination of a or b. This generalizes identities of MacMahon and Andrews. The analogous identities for three or more integers (in place of a,b) hold in certain cases.