Relations between cumulants in noncommutative probability

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.

Monotone, free, and boolean cumulants: A shuffle algebra approach

“…Monotone partitions of subsets of rns are defined similarly. We refer the reader to [1] for details on monotone partitions.…”

“…corresponds to the forest τ pπq of rooted trees encoding the nesting structure of the noncrossing partition π P N C n . See [1] for details.…”

“…, a n q :" Φpwq. We will show that Φ " exp˚pρq applied to a word w " a 1¨¨¨an P T pAq of length n can be given in terms of monotone partitions, or equivalently, in terms of non-crossing partitions weighted by inverse tree factorials [1] …”

“…[18], also for further references), the literature on the interplay between the various notions of independence and cumulants has flourished and is still flourishing, as illustrated by recent works such as [1,9,12,13,14,20].…”

“…This extends consistently to new algebraic relations among the infinitesimal characters corresponding to monotone, free, and boolean cumulants (Theorem 14). We show, among others, how to recover from this pre-Lie algebra approach the multivariate formulas describing relations between monotone, free, and boolean cumulants in terms of (irreducible) non-crossing partitions presented in reference [1].…”