The generating function of k-th power partitions is strongly Gaussian, so the asymptotic p_k(n) ~ alpha_k n^(-(3k+1)/(2k+2)) exp(beta_k n^{1/(k+1)}) follows from Hayman's theorem via mean and variance approximations.
J.: Some Gaussianity criteria for Khinchin families.J
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Power partitions and Khinchin families
The generating function of k-th power partitions is strongly Gaussian, so the asymptotic p_k(n) ~ alpha_k n^(-(3k+1)/(2k+2)) exp(beta_k n^{1/(k+1)}) follows from Hayman's theorem via mean and variance approximations.