Generalized cumulants provide a powerful framework for the analysis of non-linear statistical quantities and arise in several areas of statistics, including likelihood expansions, saddlepoint approximations, bootstrap procedures, and ratios of quadratic forms. They can be viewed as intermediate quantities between moments and cumulants, allowing the systematic computation of cumulants of polynomial functions of random variables. Despite their theoretical importance, their practical use remains limited because their evaluation requires the enumeration of complementary set partitions, a combinatorial problem that becomes increasingly difficult as the order grows.

This talk introduces a new combinatorial approach for the efficient computation of complementary set partitions. Unlike existing methods based on graph connectivity, Laplacian matrices, or symbolic algebra, the proposed algorithm exploits simple constructions involving two-block partitions. By identifying non-complementary partitions and recovering the complementary ones by set difference, the method provides a computationally efficient and scalable alternative. An implementation in R is developed, making these techniques available in a widely used statistical environment. Computational comparisons in Maple show superior performance.

The talk also extends the classical notion of generalized cumulants to settings involving repeated random variables. Using multiset subdivisions and multi-index partitions, generalized multivariate cumulants are introduced and explicit expansions in terms of multivariate cumulants are derived. Finally, unbiased estimators are obtained through k-statistics, multivariate polykays, and canonical transformations between multi-index and set partitions. The resulting framework makes generalized cumulants more accessible for modern statistical applications.