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

I. P. Goulden, Alexandru Nica, Doron Puder

Research output: Contribution to journalArticlepeer-review

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).

Original languageEnglish
Pages (from-to)983-1034
Number of pages52
JournalInternational Mathematics Research Notices
Volume2020
Issue number4
DOIs
StatePublished - 20 Feb 2020

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