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 $(pi ,ho )in extrm{NC}(n)^{2}$—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 $c_{n}pprox rac{n}{3}+rac{28}{27}$ for $no 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 $oxplus $ 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 $mathrm{lim inf}_{no infty }c_{n}^{prime }geq 0.17$. We also note that, by a direct elementary argument, one has $mathrm{lim sup}_{no infty }c_{n}^{prime }leq 0.5$. 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).
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