Abstract
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.
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Acknowledgements
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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Dedicated to my teacher, George E. Andrews, on the occasion of his 80th birthday
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The author thanks the National Security Agency for partially supporting his research program via Grant H98230-14-1-0159 during 2014–2015, when this research project commenced.
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Sills, A.V. The Combinatorics of MacMahon’s Partial Fractions. Ann. Comb. 23, 1073–1086 (2019). https://doi.org/10.1007/s00026-019-00460-9
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DOI: https://doi.org/10.1007/s00026-019-00460-9