### Abstract

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 language | English (US) |
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Pages (from-to) | 322-353 |

Number of pages | 32 |

Journal | Journal of Number Theory |

Volume | 201 |

DOIs | |

State | Published - Aug 1 2019 |

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### Keywords

- Asymptotic formula
- Restricted integer partitions

### Cite this

**A generalized Hardy-Ramanujan formula for the number of restricted integer partitions.** / Jiang, Tiefeng; Wang, Ke.

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 201, pp. 322-353. https://doi.org/10.1016/j.jnt.2019.02.006

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TY - JOUR

T1 - A generalized Hardy-Ramanujan formula for the number of restricted integer partitions

AU - Jiang, Tiefeng

AU - Wang, Ke

PY - 2019/8/1

Y1 - 2019/8/1

N2 - 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.

AB - 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.

KW - Asymptotic formula

KW - Restricted integer partitions

UR - http://www.scopus.com/inward/record.url?scp=85063497005&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063497005&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2019.02.006

DO - 10.1016/j.jnt.2019.02.006

M3 - Article

AN - SCOPUS:85063497005

VL - 201

SP - 322

EP - 353

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -