Statistics on monotonically ordered non-crossing partitions

We study some combinatorial statistics defined on the set $N{C}^{(mton)}(n)$ of monotonically ordered non-crossing partitions of {1,…,n}, and on the set $N{C}_2^{(mton)}(2n)$ 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 $N{C}^{(mton)}(n)$ and for the expectation of the area statistic on $N{C}_2^{(mton)}(2n)$ 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 $N{C}^{(mton)}(n)$'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.