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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Arithmetic of partitions and the $q$-bracket operator
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by Robert Schneider PDF
Proc. Amer. Math. Soc. 145 (2017), 1953-1968 Request permission

Abstract:

We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and find many theorems of classical number theory arise as particular cases of extremely general combinatorial structure laws. We then see that the relatively recently-defined $q$-bracket operator $\left <f\right >_q$, studied by Bloch–Okounkov, Zagier, and others for its quasimodular properties, plays a deep role in the theory of partitions, quite apart from questions of modularity. Moreover, we give an explicit formula for the coefficients of $\left <f\right >_q$ for any function $f$ defined on partitions, and, conversely, give a partition-theoretic function whose $q$-bracket is a given power series.
References
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Additional Information
  • Robert Schneider
  • Affiliation: Department of Mathematics and Computer Science, Emory University, 400 Dowman Drive, W401, Atlanta, Georgia 30322
  • ORCID: 0000-0002-4572-2344
  • Email: robert.schneider@emory.edu
  • Received by editor(s): February 23, 2016
  • Received by editor(s) in revised form: July 5, 2016
  • Published electronically: November 3, 2016
  • Communicated by: Kathrin Bringmann
  • © Copyright 2016 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 1953-1968
  • MSC (2010): Primary 05A17, 11P84; Secondary 11A25
  • DOI: https://doi.org/10.1090/proc/13361
  • MathSciNet review: 3611312