Simply generated non-crossing partitions

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.

31 pages, 11 figures

References in corpus (2)

• Recurrence of bipartite planar maps
• On the number of large triangles in the Brownian triangulation and fragmentation processes

Cited by in corpus (2)

• Scaling limits of random bipartite planar maps with a prescribed degree sequence
• Cycles in random meander systems