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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).

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