# The combinatorics of MacMahon's partial fractions

## TL;DR

This paper generalizes MacMahon's partial fractions decomposition for generating functions of partitions, offering a comprehensive combinatorial explanation and extending the classical result.

## Contribution

It introduces a generalized form of MacMahon's partial fractions decomposition with a full combinatorial interpretation.

## Key findings

- Generalization of MacMahon's partial fractions for partition generating functions
- Provides a combinatorial explanation for the decomposition
- Extends classical results to broader partition contexts

## Abstract

MacMahon showed that the generating function for partitions into at most $k$ parts can be decomposed into a partial fractions-type sum indexed by the partitions of $k$. In this present work, a generalization of MacMahon's result is given, which in turn provides a full combinatorial explanation.