Simply generated non-crossing partitions

TL;DR

This paper introduces simply generated non-crossing partitions, linking them to plane trees via a bijection, and derives limit theorems for their structure, with applications in free probability.

Contribution

It develops a new framework for simply generated non-crossing partitions and establishes a bijection with simply generated trees, enabling analysis of their asymptotic properties.

Findings

Derived limit theorems for block structures

Established a bijection with plane trees

Applied results to free probability measures

Abstract

We introduce and study the model of simply generated non-crossing partitions, which are, roughly speaking, chosen at random according to a sequence of weights. This framework encompasses the particular case of uniform non-crossing partitions with constraints on their block sizes. Our main tool is a bijection between non-crossing partitions and plane trees, which maps such simply generated non-crossing partitions into simply generated trees so that blocks of size $k$ are in correspondence with vertices of outdegree $k$. This allows us to obtain limit theorems concerning the block structure of simply generated non-crossing partitions. We apply our results in free probability by giving a simple formula relating the maximum of the support of a compactly supported probability measure on the real line in term of its free cumulants.