This paper develops new combinatorial approaches to analyze and compute special set partitions, called complementary set partitions, which are fundamental in the study of generalized cumulants. Moving away from traditional graph-based and algebraic methods, a simple and fast algorithm is proposed to list complementary set partitions based on two-block partitions, making the computation more accessible and implementable also in non-symbolic programming languages like R. Computational comparisons in Maple demonstrate the efficiency of the proposal. Additionally the notion of generalized cumulant is extended using multiset subdivisions and multi-index partitions to include scenarios with repeated variables and to address more sophisticated dependence structures. A formula is provided that expresses generalized multivariate cumulants as linear combinations of multivariate cumulants, weighted by coefficients that admit a natural combinatorial interpretation. Finally, the introduction of dummy variables and specialized multi-index partitions enables an efficient procedure for estimating generalized multivariate cumulants with a substantial reduction in data power sums involved.