Efficient Computation of Complementary Set Partitions with Applications to Generalized Cumulants

This talk presents a new combinatorial framework for the efficient computation of complementary set partitions, which play a central role in the theory of generalized cumulants. Generalized cumulants are used to express joint cumulants of polynomial functions of random variables and arise in several areas of statistics, including likelihood theory, bootstrap methods, and time series analysis. Despite their broad applicability, the practical use of generalized cumulants is often limited by the high computational complexity involved in enumerating complementary set partitions.

Most existing approaches rely on connected graph representations, Laplacian matrix calculations, or symbolic algebra systems. Although theoretically sound, these methods quickly become impractical as the number of indices increases and are difficult to implement efficiently in non-symbolic programming environments. In this talk, we propose a novel and purely combinatorial algorithm that overcomes these limitations. The key idea is to characterize all non-complementary partitions using only two-block partitions of the block index set. Complementary set partitions are then obtained by exclusion, avoiding graph-based constructions altogether.

The resulting algorithm is conceptually simple, computationally efficient, and well suited for implementation in open-source, non-symbolic languages such as R. Numerical comparisons with existing methods show a substantial reduction in computation time for moderate and large partition sizes, confirming the scalability of the proposed approach.

From a statistical perspective, the talk also introduces an extension of generalized cumulants to settings involving repeated random variables. This extension is based on multiset subdivisions and their representation via multi-index partitions, allowing the inclusion of powers of random variables and more complex dependence structures. By exploiting a suitable labeling rule and a representation of set partitions as binary vector partitions, we derive closed-form expressions for generalized multivariate cumulants in terms of products of multivariate cumulants, together with a combinatorial interpretation of the associated coefficients.