Asymptotics for a Class of Meandric Systems, via the Hasse Diagram of NC(n)

We consider the framework of closed meandric systems and its equivalent description in terms of the Hasse diagrams of the lattices of non-crossing partitions NC(n). In this equivalent description, considerations on the number of components of a random meandric system of order n translate into considerations about the distance between two random partitions in NC(n). We put into evidence a class of couples $(\pi, \rho) \in \textrm{NC}(n)^{2}$ - namely the ones where $\pi $ is conditioned to be an interval partition - for which it turns out to be tractable to study distances in the Hasse diagram. As a consequence, we observe a nontrivial class of meanders (i.e., connected meandric systems), which we call meanders with shallow top and which can be explicitly enumerated. Moreover, denoting by $c-{n}$ the expected number of components for the corresponding notion of meandric system with shallow top of order n, we find the precise asymptotic $c-{n}\approx \frac{n}{3}+\frac{28}{27}$ for $n\to \infty $. Our calculations concerning expected number of components are related to the idea of taking the derivative at t = 1 in a semigroup for the operation $\boxplus $ of free probability (but the underlying considerations are presented in a self-contained way and can be followed without assuming a free probability background). Let $c-{n}^{\prime }$ denote the expected number of components of a general, unconditioned, meandric system of order n. A variation of the methods used in the shallow-top case allows us to prove that $\mathrm{lim\ inf}-{n\to \infty }c-{n}^{\prime }/n\geq 0.17$. We also note that, by a direct elementary argument, one has $\mathrm{lim\ sup}-{n\to \infty }c-{n}^{\prime }/n\leq 0.5$. These bounds support the conjecture that $c-{n}^{\prime }$ follows a regime of constant times n (where numerical experiments suggest that the constant should be ≈ 0.23).