Limit Theorems for the Number of Summands in Integer Partitions

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.