Moments of partition statistics, Bell polynomials and Eisenstein-type series

We develop a systematic method to express generating functions for moments of combinatorial statistics in terms of partition traces. We employ an algebraic approach based on the complete Bell polynomials and their inversion formula, alongside an analytic approach via Fa\`a di Bruno's formula. Our approach can be applied to a wide class of combinatorial statistics, such as the largest part of an integer partition, the partition crank and rank, and the unimodal sequence rank.