Moments of random integer partitions

We study the limiting behaviour of the $p$th moment, that is the sum of $p$th powers of parts in a partition of a positive integer $n$ which is taken uniformly among all partitions of $n$, as $n\to\infty$ and $p\in\mathbb{R}$ is fixed. We prove that after an appropriate centring and scaling, for $p\ge 1/2$ ($p\ne 1$) the limit distribution is Gaussian, while for $p<1/2$ the limit is some infinitely divisible distribution, depending on $p$, which we describe explicitly. In particular, for $p=0$ this is the Gumbel distribution, which is well known, and for $p=-1$ the limiting distribution is connected to the Jacobi theta function.