Relations between cumulants in noncommutative probability

TL;DR

This paper explores the relationships between various types of cumulants in noncommutative probability, providing combinatorial formulas that connect classical, free, Boolean, and monotone cumulants.

Contribution

It introduces combinatorial methods to express and relate different cumulants, advancing understanding of their interconnections in noncommutative probability.

Findings

Derived formulas relating classical, free, Boolean, and monotone cumulants

Used combinatorics of heaps, pyramids, Tutte polynomials, and permutations

Partially computed coefficients for classical in terms of monotone cumulants

Abstract

We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed.