Abstract
In a series of papers, George Andrews and various coauthors successfully revitalized seemingly forgotten, powerful machinery based on MacMahon’s Ω operator to systematically compute generating functions \({\sum \nolimits }_{\lambda \in P}{z}_{1}^{{\lambda }_{1}}\ldots {z}_{n}^{{\lambda }_{n}}\) for some set P of integer partitions λ = (λ1, …, λ n ). Our goal is to geometrically prove and extend many of Andrews et al.’s theorems, by realizing a given family of partitions as the set of integer lattice points in a certain polyhedron.
Mathematics Subject Classification (2000): Primary 11P84; Secondary 05A15, 05A17, 11P21
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Notes
- 1.
After a preprint of the current article was made public, Matjaz Konvalinka and Igor Pak communicated to us that they resolved Conjecture 1 by a direct combinatorial argument.
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Acknowledgements
We thank Carla Savage for pointing out several results in the literature that were relevant to our project. This research was partially supported by the NSF through grants DMS-0810105 (Beck), DMS-0758321 (Braun), and DGE-0841164 (Le).
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Dedicated to the memory of Leon Ehrenpreis
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Beck, M., Braun, B., Le, N. (2013). Mahonian Partition Identities via Polyhedral Geometry. In: Farkas, H., Gunning, R., Knopp, M., Taylor, B. (eds) From Fourier Analysis and Number Theory to Radon Transforms and Geometry. Developments in Mathematics, vol 28. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4075-8_3
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