Statistics on monotonically ordered non-crossing partitions

TL;DR

This paper investigates combinatorial statistics on monotonically ordered non-crossing partitions, revealing a logarithmic regime in expectations and variances, and introduces a tree structure to facilitate calculations relevant to monotone probability.

Contribution

It introduces a novel tree structure on monotonically ordered non-crossing partitions and analyzes their statistical properties, highlighting differences from unordered cases.

Findings

Expectations and variances exhibit a logarithmic regime.

A tree structure simplifies combinatorial calculations.

Application to cumulants in monotone probability.

Abstract

We study some combinatorial statistics defined on the set $NC^{(mton)}(n)$ of monotonically ordered non-crossing partitions of {1,...,n}, and on the set $NC_2^{(mton)}(2n)$ of monotonically ordered non-crossing pair-partitions of {1,...,2n}. Unlike in the analogous results known for unordered non-crossing partitions, the computations of expectations and variances for natural block-counting statistics on $NC^{(mton)}(n)$ and for the expectation of the area statistic on $NC_2^{(mton)}(2n)$ turn out to yield a logarithmic regime. An important role in our study is played by a nice tree structure on the disjoint union of the $NC^{(mton)}(n)$'s, which we use to streamline our arguments. As an illustration of how these ideas can be applied to calculations of cumulants in monotone probability, we discuss some combinatorial aspects of the monotonic Poisson process.