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arXiv · 1503.09174

Simply generated non-crossing partitions

Igor Kortchemski·Cyril Marzouk
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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 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.

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Igor Kortchemski, Cyril Marzouk. 2015-03-31. Simply generated non-crossing partitions. https://doi.org/10.1017/s0963548317000050

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