The Combinatorics of MacMahon’s Partial Fractions
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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.
KeywordsPartitions Partition function Compositions Symmetric group
Mathematics Subject ClassificationPrimary 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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