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Arithmetic of partitions and the $ q$-bracket operator

Author: Robert Schneider
Journal: Proc. Amer. Math. Soc. 145 (2017), 1953-1968
MSC (2010): Primary 05A17, 11P84; Secondary 11A25
Published electronically: November 3, 2016
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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.

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Additional Information

Robert Schneider
Affiliation: Department of Mathematics and Computer Science, Emory University, 400 Dowman Drive, W401, Atlanta, Georgia 30322

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
Article copyright: © Copyright 2016 American Mathematical Society

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