The Combinatorics of MacMahon’s Partial Fractions

  • Andrew V. SillsEmail author


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


Partitions Partition function Compositions Symmetric group 

Mathematics Subject Classification

Primary 05A17 



The author thanks George Andrews for pointing out [3, p. 209, Ex. 1], which leads to the research culminating in this paper. The author thanks Matthew Katz for his interest and useful suggestions. The author particularly thanks Robert Schneider for discussions and encouragement of this project over a long period of time, and for carefully reading and offering concrete suggestions to improve earlier versions of the manuscript. Finally, the author is extremely grateful to the editor and anonymous referees for carefully reading the manuscript, catching errors, offering numerous helpful suggestions, and for their kind patience as the author prepared revisions.


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Copyright information

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Authors and Affiliations

  1. 1.Department of Mathematical SciencesGeorgia Southern UniversityStatesboroUSA

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