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 = a1 +  ⋯  + aℓ with integers 1 ≤ a1 ≤  ⋯  ≤ aℓ has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into [Formula: see text] with 1 ≤ a1 <  ⋯  < aℓ 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.