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.
Bibliographical noteFunding 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.
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- Asymptotic formula
- Restricted integer partitions