Enumerating meandric systems with large number of loops

TL;DR

This paper develops a free probability-based approach to enumerate meandric systems with many loops, providing explicit formulas, asymptotic analysis, and bounds for their generating functions.

Contribution

It introduces a novel method linking free probability to meandric system enumeration, deriving explicit formulas and asymptotics for systems with large numbers of loops.

Findings

Generating functions are rational after variable change.

Explicit formulas obtained for systems with up to 6 loops.

Asymptotic behavior of meandric numbers for general r.

Abstract

We investigate meandric systems with a large number of loops using tools inspired by free probability. For any fixed integer $r$, we express the generating function of meandric systems on $2n$ points with $n-r$ loops in terms of a finite (the size depends on $r$) subclass of irreducible meandric systems, via the moment-cumulant formula from free probability theory. We show that the generating function, after an appropriate change of variable, is a rational function, and we bound its degree. Exact expressions for the generating functions are obtained for $r \leq 6$, as well as the asymptotic behavior of the meandric numbers for general $r$.