In this talk, I will introduce a new way of sampling random non-crossing partitions which takes into account their nesting structure. Specifically, the weight of a non-crossing partition corresponds to the number of its so-called monotonic orderings; this structure has appeared in the literature on monotone cumulants, and is somewhat analogous to the well-known branching structure of Young diagrams. Then, I will show how various statistics on non-crossing partitions, pairings, and related combinatorial objects behave asymptotically with respect to this new probability measure. Based on joint work with Natasha Blitvic, Thomas Bray, and Alexandru Nica.