Symmetric functions and the explicit moment problem for abelian groups

Recently, Sawin and Wood (arXiv:math/2210.06279) proved a formula for the distribution of a random abelian group $G$ in terms of its $H$-moments $\mathbb{E}[\mathrm{\#}\mathrm{Sur}(G,H)]$. We show that properties of Macdonald polynomials yield an alternate proof.