Enumerating meandric systems with large number of loops

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 $r\le 6$, as well as the asymptotic behavior of the meandric numbers for general $r$.