Power partitions and Khinchin families

TL;DR

This paper proves that the generating function for partitions into k-th powers exhibits strong Gaussian behavior, leading to an explicit asymptotic formula for the partition count using probabilistic and analytic techniques.

Contribution

It establishes strong Gaussianity of the generating function for k-th power partitions and derives an explicit Hardy-Ramanujan type asymptotic formula.

Findings

The generating function is strongly Gaussian in the sense of Baez-Duarte.

An explicit asymptotic formula for p_k(n) is obtained for large n.

The proof combines Gaussianity criteria with bounds on the generating function.

Abstract

We prove that the generating function of partitions into $k$-th powers is strongly Gaussian in the sense of B\'aez-Duarte. Within the probabilistic framework of Khinchin families, the Hardy--Ramanujan asymptotic formula for the number~$p_k(n)$ of partitions of~$n$ into $k$-th powers reads \[ p_k(n) \sim \frac{\alpha_k}{n^{(3k+1)/(2k+2)}} \exp\bigl(\beta_k\, n^{1/(k+1)}\bigr), \qquad n \to \infty, \] where $\alpha_k$ and $\beta_k$ are explicit constants depending only on~$k$, then follows directly from Hayman's asymptotic formula for strongly Gaussian power series. The proof of strong Gaussianity combines a Gaussianity criterion for Khinchin families with bounds of Tenenbaum, Wu and Li on the generating function; the asymptotic formula is recovered by computing asymptotic approximations of the mean and variance of the associated family.