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The Ramanujan Journal

, Volume 47, Issue 2, pp 339–381 | Cite as

Partitions and Sylvester waves

  • Cormac O’Sullivan
Article
  • 132 Downloads

Abstract

The restricted partition function \(p_N(n)\) counts the partitions of the integer n into at most N parts. In the nineteenth century, Sylvester described these partitions as a sum of waves. We give detailed descriptions of these waves and, for the first time, show the asymptotics of the initial waves as N and n both go to infinity at about the same rate. This allows us to see when the initial waves are a good approximation to \(p_N(n)\) in this situation. Our proofs employ the saddle-point method of Perron and the dilogarithm.

Keywords

Restricted partitions Sylvester waves Asymptotics Saddle-point method 

Mathematics Subject Classification

11P82 41A60 

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsThe Cuny Graduate CenterNew YorkUSA

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