Simply generated non-crossing partitions

Igor Kortchemski (1, 2) , Cyril Marzouk (3)

1 CNRS - Centre National de la Recherche Scientifique
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 UZH - Universität Zürich [Zürich] = University of Zurich

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 out-degree $k$. 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 terms of its free cumulants.

Domaines: Probabilités [math.PR]

Igor Kortchemski, Cyril Marzouk. Simply generated non-crossing partitions. Combinatorics, Probability and Computing, 2017, 26 (4), pp.560-592. ⟨10.1017/S0963548317000050⟩. ⟨hal-01199288⟩