Asymptotic Distributions for Block Statistics on Non-crossing Partitions

Abstract

The set of non-crossing partitions was first studied by Kreweras in 1972 and was known to play an important role in combinatorics, geometric group theory, and free probability. In particular, it has a natural embedding into the symmetric group, and there is an extensive literature on the asymptotic cycle structures of random permutations. This motivates our study on analogous results regarding the asymptotic block structure of random non-crossing partitions.

We first investigate an analogous result of the asymptotic distribution for the total number of cycles of random permutations due to Goncharov in 1940's: Goncharov showed that the total number of cycles in a random permutation is asymptotically normally distributed with mean log(n) and variance log(n). As a analog of this result, we show that the total number of blocks in a random non-crossing partition is asymptotically normally distributed with mean n/2 and variance n/8.

We also investigate the outer blocks, which arise naturally from non-crossing partitions and has many connections in combinatorics and free probability. It is a surprising result that among many blocks of non-crossing partitions, the expected number of outer blocks is asymptotically 3. We further computed the asymptotic distribution for the total number of blocks, which is a shifted negative binomial distribution.