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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 points with loops in terms of a finite (the size depends on ) 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 , as well as the asymptotic behavior of the meandric numbers for general .
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. 607–640DOI 10.4171/AIHPD/80