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Simply Generated Non-Crossing Partitions

Published online by Cambridge University Press:  28 March 2017

IGOR KORTCHEMSKI  and
CYRIL MARZOUK
IGOR KORTCHEMSKI
Affiliation:
CNRS, CMAP, École Polytechnique, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau CEDEX, France (e-mail: igor.kortchemski@normalesup.org)
CYRIL MARZOUK
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland (e-mail: cyril.marzouk@math.uzh.ch)
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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 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.

Keywords

Primary 05C8060C05Secondary 05C0560J80

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 4 , July 2017 , pp. 560 - 592
Copyright
Copyright © Cambridge University Press 2017 

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