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

• Published in: International mathematics research notices 2020-02, Vol.2020 (4), p.983-1034
• Main Authors: Goulden, I P, Nica, Alexandru, Puder, Doron
• Format: Article
• Language: English
• ISSN: 1073-7928; 1687-0247
• DOI: 10.1093/imrn/rny044

Abstract 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).