A generalized Hardy-Ramanujan formula for the number of restricted integer partitions

Tiefeng Jiang, Ke Wang

Research output: Contribution to journalArticlepeer-review


We derive an asymptotic formula for p n (N,M), the number of partitions of integer n with part size at most N and length at most M. We consider both N and M are comparable to n. This is an extension of the classical Hardy-Ramanujan formula and Szekeres' formula. The proof relies on the saddle point method.

Original languageEnglish (US)
Pages (from-to)322-353
Number of pages32
JournalJournal of Number Theory
StatePublished - Aug 2019

Bibliographical note

Funding Information:
The research of Tiefeng Jiang is supported in part by NSF Grant DMS-1209166 and DMS-1406279.Ke Wang is supported by HKUST Initiation Grant IGN16SC05.

Publisher Copyright:
© 2019 Elsevier Inc.


  • Asymptotic formula
  • Restricted integer partitions

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