Journal of Combinatorial Theory, Series A

Volume 96, Issue 1, October 2001, Pages 89-126

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Regular Article

Limit Theorems for the Number of Summands in Integer Partitions ☆

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Hsien-Kuei Hwang
Institute of Statistical Science, Academia Sinica, Taipei, 115, Taiwan

Received 8 November 2000, Available online 1 March 2002

doi:10.1006/jcta.2000.3170

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Abstract

Central and local limit theorems are derived for the number of distinct summands in integer partitions, with or without repetitions, under a general scheme essentially due to Meinardus. The local limit theorems are of the form of Cramér-type large deviations and are proved by Mellin transform and the two-dimensional saddle-point method. Applications of these results include partitions into positive integers, into powers of integers, into integers [jβ], β>1, into aj+b, etc.

Keywords

  • integer partitions;
  • central and local limit theorems;
  • large deviations;
  • Meinardus's scheme;
  • Mellin transform;
  • Lerch's zeta function;
  • saddle-point method
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Communicated by Andrew, Odlyzko

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