Partitions into distinct parts with bounded largest part

Abstract

We prove an asymptotic formula for the number of partitions of n into distinct parts where the largest part is at most \(t\sqrt{n}\) for fixed \(t \in {\mathbb {R}}\). Our method follows a probabilistic approach of Romik, who gave a simpler proof of Szekeres’ asymptotic formula for distinct parts partitions when instead the number of parts is bounded by \(t\sqrt{n}\). Although equivalent to a circle method/saddle-point method calculation, the probabilistic approach motivates some of the more technical steps and even predicts the shape of the asymptotic formula, to some degree.

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Correspondence to Walter Bridges.

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Bridges, W. Partitions into distinct parts with bounded largest part. Res. number theory 6, 40 (2020). https://doi.org/10.1007/s40993-020-00219-9

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Keywords

  • Partitions
  • Asymptotic analysis
  • Circle method
  • Probability

Mathematics Subject Classification

  • 05A17
  • 11P82