Boolean, Free, and Classical Cumulants as Tree Enumerations

TL;DR

This paper extends the combinatorial framework of cumulants using weighted troupes to multivariate cases and provides new insights into Boolean cumulants and specific distributions.

Contribution

It generalizes the troupe-based combinatorial description of cumulants to multivariate and weighted cases, also addressing Boolean cumulants and troupe transforms.

Findings

Generalization of troupes to weighted troupes for multivariate cumulants

Combinatorial description of Boolean cumulants

Explicit distributions matching specific weighted troupes

Abstract

Defant found that the relationship between a sequence of (univariate) classical cumulants and the corresponding sequence of (univariate) free cumulants can be described combinatorially in terms of families of binary plane trees called troupes. Using a generalization of troupes that we call weighted troupes, we generalize this result to allow for multivariate cumulants. Our result also gives a combinatorial description of the corresponding Boolean cumulants. This allows us to answer a question of Defant regarding his troupe transform. We also provide explicit distributions whose cumulants correspond to some specific weighted troupes.