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

I. P. Goulden, Alexandru Nica, Doron Puder

Published 2017-08-17Version 1

We consider closed meandric systems, and their 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 focus on a class of couples $(\pi,\rho)\in NC(n)^2$ -- 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 non-trivial class of meanders (i.e. connected meandric systems), which we call "meanders with shallow top", and which can be explicitly enumerated. Moreover, the expected number of components for a random "meandric system with shallow top", is asymptotically $(9n+28)/27$. A variation of the methods used in the shallow-top case yields non-trivial bounds on the expected number of components of a general (unconditioned) random meandric system of order $n$. We show this expected number falls inside $(0.17n,0.51n)$ for large enough $n$. Our calculations here 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). Another variation of these methods goes by fixing a "base-point" $\lambda_{n}$, where $\lambda_{n}$ is an interval partition in $NC(n)$, and by focusing on distances in the Hasse diagram of $NC(n)$ which are measured from $\lambda_{n}$. We illustrate this by determining precise formulas for the average distance to $\lambda_{n}$ and for the cardinality of $\{\rho\in NC(n)\mid\rho$ at maximal distance from $\lambda_{n}\}$ in the case when $n$ is even and $\lambda_{n}$ is the partition with blocks $\{1,2\},\{3,4\},\ldots,\{n-1,n\}$.