Abstract
We explore partitions that lie in the intersection of several sets of classical interest: partitions with parts indivisible by m, appearing fewer than m times, or differing by less than m. We find results on their behavior and generating functions: more results for those simultaneously regular and distinct, fewest for those distinct and flat. We offer some conjectures in the area.
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References
D. Stockhofe, Bijektive Abbildungen auf der Menge der Partitionen einer Naturlichen Zahl. Bayreuth. Math. Schr. 10, 1–59 (1982)
W. Keith, Ranks of partitions and Durfee symbols. Ph.D. Thesis, Pennsylvania State University, (June 2007). http://etda.libraries.psu.edu/theses/approved/WorldWideIndex/ETD-2026/index.html
S. Treneer, Congruences for the coefficients of weakly holomorphic modular forms. Proc. Lond. Math. Soc. 93, 304–324 (2006)
Ø. Rødseth, Dissections of the generating functions of \(q(n)\) and \(q_0(n)\). Arbok University Bergen Mat. Nat. 1969, 12 (1970), 3–12. MR0434959 (55:7922)
E.X.W. Xia, O.X.M. Yao, Analogues of Ramanujan’s partition identities. Ramanujan J. 31, 373–396 (2013). https://doi.org/10.1007/s11139-012-9439-x
R.E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras. Invent. Math. 109, 405–444 (1992)
The On-Line Encyclopedia of Integer Sequences, published electronically at https://oeis.org Dec 2016
McKay-Thompson series of class 54c for the Monster group. https://oeis.org/A112194
Stanley, R. Enumerative Combinatorics, vol. 1, Cambridge Studies in advanced Mathematics vol. 49, eds. by Fulton, Garling, Ribet, and Walters (Cambridge University Press, NY, 1997)
Acknowledgements
The author warmly thanks Melvyn B. Nathanson and all other organizers and staffers of CANT 2016 for the opportunity to speak and the production of this proceedings volume, and for the lively and interesting discussions and problem sessions which surround the presentations at the conference.
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Keith, W.J. (2017). Partitions into Parts Simultaneously Regular, Distinct, And/or Flat. In: Nathanson, M. (eds) Combinatorial and Additive Number Theory II. CANT CANT 2015 2016. Springer Proceedings in Mathematics & Statistics, vol 220. Springer, Cham. https://doi.org/10.1007/978-3-319-68032-3_10
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DOI: https://doi.org/10.1007/978-3-319-68032-3_10
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