A central limit theorem for integer partitions into small powers.

The study of the well-known partition function p ( n ) counting the number of solutions to n = a 1 + ⋯ + a ℓ with integers 1 ≤ a 1 ≤ ⋯ ≤ a ℓ has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into n = a 1 α + ⋯ + a ℓ α with 1 ≤ a 1 < ⋯ < a ℓ and some fixed 0 < α < 1 . In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle-point method.