# Enumerating meandric systems with large number of loops

### Motohisa Fukuda

Yamagata University, Japan### Ion Nechita

TU München, Germany and Université de Toulouse, France

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## 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 $n \to r$ 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 $r$.

## Cite this article

Motohisa Fukuda, Ion Nechita, Enumerating meandric systems with large number of loops. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 6 (2019), no. 4, pp. 607–640

DOI 10.4171/AIHPD/80