Efficient computation of complementary set partitions, with applications to an extension and estimation of generalized cumulants

TL;DR

This paper introduces a fast combinatorial algorithm for computing complementary set partitions and extends generalized cumulants to more complex dependence structures, improving computational efficiency and applicability.

Contribution

A new simple algorithm for listing complementary set partitions and an extension of generalized cumulants using multiset subdivisions and multi-index partitions.

Findings

Algorithm outperforms traditional methods in computational efficiency.

Extended generalized cumulants to include repeated variables and complex dependencies.

Significant reduction in data power sums for cumulant estimation.

Abstract

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.